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This paper is devoted to studying the regularity properties for the new maximal operator M φ and the fractional new maximal operator M φ , β in the local case. Some new pointwise gradient estimates of M φ , Ω and M φ , β , Ω are given. Moreover, the boundedness of M φ , Ω and M φ , β , Ω on Sobolev space is established. As applications, we also obtain the bounds of the above operators on Sobolev space with zero boundary values. [ABSTRACT FROM AUTHOR]

In this paper, mixed moduli of smoothness of functions of two variables are studied. We prove Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in different metrics. Estimates for the mixed moduli of smoothness of the derivative of a function are also obtained in terms of the mixed moduli of smoothness of the function itself. [ABSTRACT FROM AUTHOR]

In this paper we study the pointwise convergence of Landau type Schrödinger operators. We prove the non-tangential convergence for such operators on weighted Lebesgue spaces, which extends a known result by Yuan, Zhao and Zheng. Moreover, we also study the pointwise convergence of Landau type Schrödinger operators along sequences. Our results are established by proving the corresponding maximal estimates. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical Schrödinger–Poisson system involving p-Laplacian in R 3 of the form: - Δ p u - ϕ | u | p - 2 u = λ | u | p - 2 u + μ | u | q - 2 u + | u | p ∗ - 2 u in R 3 , - Δ ϕ = | u | p in R 3 , and prescribed mass ∫ R 3 | u | p d x = a p , where Δ p u = div (| ∇ u | p - 2 ∇ u) is the p-Laplacian operator, p ∈ (1 2 , 3) , μ , a > 0 , λ ∈ R , q ∈ (p , p ∗) and p ∗ : = 3 p 3 - p . For the L p -subcritical case, we show the existence of multiple normalized solutions by using the truncation technique and the genus theory. For the L p -supercritical case, we obtain a couple of normalized solutions by developing the auxiliary functional. For both cases, in order to overcome the loss of compactness of the energy functional due to the critical growth, the concentration–compactness principle is needed to overcome this difficulty. In a sense, we generalize some of the previous results. [ABSTRACT FROM AUTHOR]

Our main goal is to investigate supercritical Hardy–Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term. [ABSTRACT FROM AUTHOR]

In this paper, the existence of at least one solution to the problem (P) \[ \begin{cases} -\mathcal{L}_Mu=\lambda f(u)\,{\rm e}^{\alpha|u|^{M'}}+\mu|u|^{p-2}u & \quad {\rm in}\ \Omega\\ u=0 & \quad {\rm on}\ \partial\Omega \end{cases} \] { − L M u = λf (u) e α | u | M ′ + μ | u | p − 2 u in Ω u = 0 on ∂Ω is established, where $ \mathcal {L}_Mu:=\operatorname {div}_{\mathbb {G}}\cdot (|\nabla _{\mathbb {G}}u|^{M-2}\nabla _{\mathbb {G}}u) $ L M u := div G ⋅ (| ∇ G u | M − 2 ∇ G u) is the M-sub-Laplacian, $ 2\leq p{ 2 ≤ p < M , $ \alpha {>}0 $ α > 0 , $ M'=\frac {M}{M-1} $ M ′ = M M − 1 , Ω is a bounded subset of the N-dimensional stratified Lie group $ \mathbb {G} $ G with $ N\geq ~3 $ N ≥ 3 (N = 1, 2 has been avoided since they do not produce a purely stratified Lie group) and M is the homogeneous dimension of $ \mathbb {G} $ G . We will also show that in fact, the problem admits infinitely many solutions with some restrictions on the parameters involved. [ABSTRACT FROM AUTHOR]

Set g = (g 0 , g 1 , ⋯ , g n - 1) with g i ∈ H (C) for i = 0 , 1 , ⋯ , n - 1 and let T g (n) be the generalized Volterra-type operators on H (C) , which is represented as T g (n) f = I n (f g 0 + f ′ g 1 + ⋯ + f (n - 1) g n - 1) , where I denotes the integration operator (I f) (z) = ∫ 0 z f (w) d w , and I n is the nth iteration of I. This operator is a generalization of the operator that was introduced by Chalmoukis in Ref. [1]. In this paper, we study the boundedness and compactness of the operators T g (n) acting on Fock spaces to another. As a consequence of these characterizations, we obtain conditions for certain linear differential equations to have solutions in Fock spaces. Then, we study the same properties, boundedness and compactness, of the following linear combination of weighted composition–differentiation operators: let u = (u 0 , ⋯ , u n) with u k ∈ H (C) for 0 ≤ k ≤ n and φ ∈ H (C). The linear combination of weighted composition–differentiation operators is defined by L u , φ (n) = ∑ i = 0 n u i C φ (i) , where u C φ (i) f = u · f (i) ∘ φ. Our approach involves the study of Sobolev Carleson measures for classical Fock spaces. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on R n to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions. [ABSTRACT FROM AUTHOR]

Given an elliptic operator L = - div (A ∇ ·) subject to mixed boundary conditions on an open subset of R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets. [ABSTRACT FROM AUTHOR]

We are concerned with the existence and mass collapse of standing waves with prescribed mass for the Schrödinger equation with general potential and nonlinearities. For local nonlinearities, this equation arises in the theory of Bose–Einstein condensates. For nonlocal nonlinearities, this equation is the Choquard euqation, which appears in the quantum theory of a polaron at rest. We used a unified approach (local minizing method) to study the existence of standing waves for the local, nonlocal and dipolar type cases. And then we established the mass collapse result. [ABSTRACT FROM AUTHOR]

This article provides a revised version of some existing results in the literature for the quaternion Fourier transform (QFT) and quaternion wavelet transforms. The inner-product relation and its consequent formula for the continuous quaternion wavelet transform (CQWT) are derived in L p (R 2 ; H) space under the assumption that the admissible wavelet is complex-valued and has a real QFT. Furthermore, the characterization of quaternion Sobolev spaces H s (R 2 ; H) and W m , p (Ω ; H) , weighted quaternion Sobolev space W k m , p (Ω ; H) and generalized quaternion Sobolev space H w ω (R 2 ; H) , quaternion Besov space by means of the CQWT is presented. The CQWT is analysed within these function and distribution spaces, yielding novel findings regarding continuity and boundedness. [ABSTRACT FROM AUTHOR]

We extend the range of parameters associated with the Gagliardo–Nirenberg interpolation inequalities in the fractional Coulomb–Sobolev spaces for radial functions. We also study the optimality of this newly extended range of parameters. [ABSTRACT FROM AUTHOR]

In the present paper, we concern on discussing some results of the fractional Laplacian with respect to a given function μ. In addition, we also discuss the Nemytskii operators Ψ ↦ | Ψ | {\Psi\mapsto|\Psi|} and Ψ ↦ Ψ ± {\Psi\mapsto\Psi^{\pm}} in μ-fractional Sobolev–Slobodetskii space. [ABSTRACT FROM AUTHOR]

We establish a pointwise property for homogeneous fractional Sobolev spaces in domains with non-empty boundary, extending a similar result of Koskela–Yang–Zhou. We use this to show that a conformal map from the unit disk onto a simply connected planar domain induces a bounded composition operator from the borderline homogeneous fractional Sobolev space of the domain into the corresponding space of the unit disk. [ABSTRACT FROM AUTHOR]

We study embeddings between generalised Triebel–Lizorkin–Morrey spaces E φ , p , q s (R d) and within the scales of further generalised Morrey smoothness spaces like N φ , p , q s (R d) , Bp,qs,φ(ℝd) and Fp,qs,φ(ℝd). The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale N φ , p , q s (R d) were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces E φ , p , q s (R d) . Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies' wavelets. Then we prove necessary and sufficient conditions for the embedding E φ 1 , p 1 , q 1 s 1 (R d) ↪ E φ 2 , p 2 , q 2 s 2 (R d) . We can also provide some almost final answer to the question when E φ , p , q s (R d) is embedded into C(ℝd), complementing our recent findings in case of N φ , p , q s (R d) . [ABSTRACT FROM AUTHOR]

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces Bps(ℝn) = Bp,ps(ℝn), 1 ≤ p ≤ ∞, and between Sobolev spaces Hps(ℝn), 1 < p < ∞. In contrast to the paper H. Triebel, Mapping properties of Fourier transforms. Z. Anal. Anwend. 41 (2022), 133–152, based mainly on embeddings between related weighted spaces, we rely on wavelet expansions, duality and interpolation of corresponding (unweighted) spaces, and (appropriately extended) Hausdorff-Young inequalities. The degree of compactness will be measured in terms of entropy numbers and approximation numbers, now using the symbiotic relationship to weighted spaces. [ABSTRACT FROM AUTHOR]

Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. Lastly, we apply our results to obtain Sobolev inequalities for non-regular Riemannian metrics on geometric exterior domains. [ABSTRACT FROM AUTHOR]

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 and 1 < p ≤ n + 2 n - 2 , there exists a best constant S n , p , λ (B n) > 0 such that S n , p , λ (B n) ∫ B n | u | p + 1 d v B n 2 p + 1 ≤ ∫ B n | ∇ B n u | 2 - λ u 2 d v B n , holds for all u ∈ C c ∞ (B n) , and λ ≤ (n - 1) 2 4 , where (n - 1) 2 4 is the bottom of the L 2 -spectrum of - Δ B n. It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635–671, 2008) that under appropriate assumptions on n, p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S n , p , λ (B n). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in R n by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18–24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780–6797, 2018) to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744–789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity. [ABSTRACT FROM AUTHOR]

We present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation. [ABSTRACT FROM AUTHOR]

We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear nonlocal integro-differential p-Lévy operators. A prototypical example of integro-differential p-Lévy operators is the well-known fractional p-Laplace operator. Our main focus is on nonlinear integro-differential equations in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional p-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge the gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with p-Laplacian are strong limits of the nonlocal ones. [ABSTRACT FROM AUTHOR]

For an arbitrary subset E ⊂ R n , we introduce and study the three vanishing subspaces of the Hölder space C ˙ 0 , ω (E) consisting of those functions for which the ratio | f (x) - f (y) | / ω (| x - y |) vanishes, when (1) | x - y | → 0 , (2) | x - y | → ∞ or (3) min (| x | , | y |) → ∞ . We prove that the Whitney extension operator maps each of these vanishing subspaces from E to the corresponding vanishing spaces defined on the whole ambient space R n . In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions C ˙ 0 , ω (E) by Lipschitz and boundedly supported functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a new nonlocal Q-Laplacian equation on the Heisenberg group in the smooth bounded domain. The nonlinear term f (ξ , u) has exponential growth behaving like exp (β | t | Q Q - 1 ) as | t | → ∞ with some β > 0 . Existence of solutions are obtained by an application of the mountain pass theorem and Ekeland's variational principle. Moreover, our results are new even on the Euclidean case. [ABSTRACT FROM AUTHOR]

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of R n , possibly non closed. We also establish density results for W ∇ 1 , p (M) , M bounded subanalytic manifold, which is the space of the L p tangent vector fields v on M for which ∇ v is L p , where ∇ is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the p-Laplace equation, for p ∈ [ 1 , ∞) large. [ABSTRACT FROM AUTHOR]

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from H α (R , U) to H β (R , U) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail. [ABSTRACT FROM AUTHOR]

We study properties of the boundary trace operator on the Sobolev space W 1 1 (Ω) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator T r : W 1 1 (Ω K) → X (Ω K) , where Ω K is von Koch's snowflake and X (Ω K) is a trace space with the quotient norm. Since Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator S : X (Ω K) → W 1 1 (Ω K) such that T r ∘ S = I d X (Ω K) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain Ω with regular boundary, which explains Banach space geometry cause for this phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a class of Besov spaces via the heat semigroup on Carnot groups, and provide some basic properties about these spaces. Especially, we reveal the relationship between the Besov space and the known Sobolev space on Carnot groups. Then some Sobolev type inequalities for Besov spaces are established. Finally, we introduce the Besov capacity associated with Besov spaces and investigate some measure theories and geometric properties for this capacity. Furthermore, several applications of the Besov capacity are also obtained. [ABSTRACT FROM AUTHOR]

In this investigation, we study the existence of weak solution for a class of nonhomogeneous anisotropic elliptic problem. These problems involve the anisotropic operators with variable exponents. Based on the topological degree theory concerning a specific subset of demicontinuous operators under generalized (S +) and the theory of anisotropic Sobolev spaces with variable exponent, we demonstrate the existence of weak solution for this problem. Our findings are applicable to problems with no-flux boundary conditions, and they extend and generalize several results previously reported in the literature. [ABSTRACT FROM AUTHOR]

For functions f ∈ L 2 R d , w l (x) d x with w l is a weight function invariant under the action of an associated Weyl group, we give necessary and sufficient conditions for f to belong to the generalized Lipschitz and Besov spaces in terms of the behaviour of their Dunkl transforms f ^ setting by means of kth-order moduli of continuity. [ABSTRACT FROM AUTHOR]

The purpose of this article is to prove the existence and uniqueness of weak solutions to the following obstacle problem of p-Laplace-type: ∫ Ω σ 1 (z , D u - F (u)) : D (v - u) + σ 2 (z , D u) : (v - u) + u | u | p - 2 , v - u d z ≥ 0 , with data belonging to the dual of Sobolev spaces. The main result is demonstrated by means of Kinderlehrer and Stampacchia’s Theorem and Young’s measure theory. [ABSTRACT FROM AUTHOR]

We investigate the τ -quantizations and Cohen’s class distributions of a suitable class of trace-class operators, called Feichtinger’s operators, and show that it is a convenient substitute for the class of Schwartz operators. Many well-known concepts and results for functions in time-frequency analysis have an operator-analog in our setting, e.g. that Cohen’s classes are convolutions of Wigner functions with distributions or characterization of the class of Schwartz operators as an intersection of weighted variants of the class of Feichtinger operators. [ABSTRACT FROM AUTHOR]

This paper is devoted in investigations concerning the study of the coupled potential operator J s α , β and corresponding L p -Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space S α , β is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The L p norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator J s α , β is defined as a pseudo-differential operator related with a precise symbol. The operator J s α , β is extended to a space of distributions. An L p -Sobolev boundedness result for the operator J s α , β is shown. The spaces H p m , α , β and H p m , α , β introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new space that generalizes the ϕ -Hilfer space with the ξ (·) -Laplacian operator, denoted (ϕ , ξ (·)) -HFDS. We refer to this new space as the ϕ -fractional space with anisotropic ξ → (·) -Laplacian operator, abbreviated as (ϕ , ξ → (·)) -HFDAS. We prove that (ϕ , ξ → (·)) -HFDAS is a separable, and reflexive Banach space. Furthermore, we extend some well-known properties and embedding results of the (ϕ , ξ (·)) -HFDS space to (ϕ , ξ → (·)) -HFDAS. Moreover, we illustrate an application of (ϕ , ξ → (·)) -HFDAS by solving a differential equation via variational methods. [ABSTRACT FROM AUTHOR]

Circular prolate spheroidal wave functions (CPSWFs) called also Hankel prolate denoted by φ n , c (α + p / 2) , c > 0 , p ∈ N and α ≥ - 1 / 2 , are the radial part of the classical band limited and best essentially time functions defined on L 2 (B d) , where B d is the unit Ball of R d , d = p + 2 ≥ 2 , introduced by D. Slepian in the 60 s. The main issue of this work is to establish the L 2 convergence quality of the truncated error given by the series expansion of a function f ∈ L 2 [ 0 , 1 ] in CPSWFs basis, then to extend the previous challenge in the L p norm, 4 / 3 < p < 4 . In the meantime, we will give two uniform approximations of the previous system by Bessel function of the first kind and a modified Jacobi polynomial which will be the important step to avoid all obstacles. Let recall that the Hankel prolate covers the classical PSWFs (α = ± 1 / 2 , p = 0) for which corresponding results have been established previously by many authors. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the elliptic problem: $$\begin{align*} (-\Delta_p)^su+|u|^{p-2}u&=\mu f(x,u)+\lambda u^{-\gamma} \textrm{ in } \Omega,\\ u&>0 \textrm{ in } \Omega,\\ \mathcal{N}_p^su+\alpha(x)|u|^{p-2}u&=0 \textrm{ in } \mathbb{R}^N\setminus\Omega, \end{align*} $$ (− Δ p) s u + | u | p − 2 u = μf (x , u) + λ u − γ in Ω , u > 0 in Ω , N p s u + α (x) | u | p − 2 u = 0 in R N ∖ Ω , under some conditions on the nonlinearity f. Here $ 2\leq p \leq p_s^{*} $ 2 ≤ p ≤ p s ∗ , $ \mu, \lambda >0 $ μ , λ > 0 , $ 0 0 < s , γ < 1 , $ \alpha \in L^{\infty }(\mathbb {R}^N\setminus \Omega) $ α ∈ L ∞ (R N ∖ Ω). We will prove the existence of at least one solution using the variational principle and the Mountain pass theorem. [ABSTRACT FROM AUTHOR]

An RD-space 푀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 푀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé's class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multiplicity results for singular generalized Kirchhoff two-phase fractional differential equation problems with a p-Laplacian operator with parameters and a nonlinear term exhibiting subcritical growth. Under broad assumptions on the data, we establish the existence of at least two weak solutions with different energy signs. Our approach is based on the fibering method, which specifically uses the Nehari manifold. It is noteworthy that our analysis includes both non-degenerate and degenerate Kirchhoff cases. These results contribute to the advancement of knowledge in the field of Kirchhoff double-phase problems within ψ -Hilfer generalized fractional differential equations with singular nonlinearities. [ABSTRACT FROM AUTHOR]

We consider a generalized singular capillarity problem driven by a fractional Hilfer derivative with respect to a function, with Dirichlet boundary conditions. The nonlinearity of the problem generally exhibits some singular characteristics and is characterized by a variable exponent function, which displays critical behavior at infinity. Using the combination of the Nehari manifold method with a variational approach on fractional spaces in the sense of Hilfer, we prove the existence and multiplicity of positive solutions to such a problem provided that the parameters that appear in the problem satisfy some appropriate conditions. Our main results are novel, and their investigation will enhance the scope of the literature on singular coupled systems involving generalized fractional derivatives. [ABSTRACT FROM AUTHOR]

In this study, we obtain the weighted Hardy-Adams inequality of any even dimension n≥4n\ge 4. Namely, for u∈C0∞(Bn)u\in {C}_{0}^{\infty }\left({{\mathbb{B}}}^{n}) with ∫Bn∣∇n2u∣2dx−∏k=1n⁄2(2k−1)2∫Bnu2(1−∣x∣2)ndx≤1,\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{| {\nabla }^{\frac{n}{2}}u| }^{2}{\rm{d}}x-\mathop{\prod }\limits_{k=1}^{n/2}{\left(2k-1)}^{2}\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}\frac{{u}^{2}}{{\left(1-{| x| }^{2})}^{n}}{\rm{d}}x\le 1, then the following Hardy-Adams inequalities hold: ∫Bnexp22−n−ϑβ0n2,n(uφϑ)2(1−∣x∣2)2ϑ+n−4dx≤Cn,ϑ,ϑ∈3−n2,0\begin{array}{r}\mathop{\displaystyle \int }\limits_{{{\mathbb{B}}}^{n}}\exp \left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(\frac{3-n}{2},0\right]\end{array} and ∫BnΦ222−n−ϑβ0n2,n(uφϑ)2(1−∣x∣2)2ϑ+n−4dx≤Cn,ϑ,ϑ∈(2−n,3−n2],\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{\Phi }_{2}\left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(2-n,\frac{3-n}{2}\right], where Φ2(t)=et−∑1j=0tjj!{\Phi }_{2}\left(t)={e}^{t}-\mathop{\sum ^{1}}\limits_{j=0}\frac{{t}^{j}}{j\!}, φϑ(x)=21−∣x∣2n+ϑ−22{\varphi }_{{\vartheta }}\left(x)={\left(,\frac{2}{1-{| x| }^{2}}\right)}^{\tfrac{n+{\vartheta }-2}{2}} and Cn,ϑ{C}_{n,{\vartheta }} is a positive constant independent of uu. The crucial method in this article is borrowed from Lu and Yang [Adv. Math. 319 (2017)] and Li, Lu and Yang [Adv. Math. 333 (2018)]. We also apply set level method which was first developed by Lam and Lu [J. Diff. Equa. 255 (2013)]. The new ingredient here is to utilize the relationship between ΔH{\Delta }_{{\mathbb{H}}} and the Laplace-Beltrami operator Δϑ{\Delta }_{{\vartheta }}, where Δϑ{\Delta }_{{\vartheta }} is developed by Liu and Peng [Indiana Univ. Math. J. 3 (2009), 1457–1492]. Furthermore, the Hardy-Adams inequality in this article is the weighted version of the result of Li et al. [Adv. Math. 333 (2018), 350–385].

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.

We investigate some of the effects of the lack of compactness in the critical Folland–Stein–Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi’s Γ-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 1984, 4, 511–517]. [ABSTRACT FROM AUTHOR]

This study explores the L p {L^{p}} -boundedness and compactness characteristics of SG pseudo-differential operators within the Sobolev space framework. We analyze different attributes of minimal-maximal pseudo-differential operators using the SG symbol σ ⁢ ( x , λ ) {\sigma(x,\lambda)} within the class S m 1 , m 2 {S^{m_{1},m_{2}}} , where m 1 {m_{1}} and m 2 {m_{2}} are real numbers. The investigation delves into various aspects of these operators, and we derive a weak solution for the SG pseudo-differential equation by employing the Dunkl transform as part of the aforementioned theoretical framework. [ABSTRACT FROM AUTHOR]

Let B be a unit ball in ℝ2, W 0 1 , 2 (B) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely, sup u ∈ W 0 1 , 2 (B) , ∫ B | ∇ u | 2 d x ≤ 1 ∫ B | x | 2 ϵ e 4 π (1 + ϵ) u 2 d x can be attained by some radially symmetric function u ϵ ∈ W 0 1 , 2 (B) with ∫ B | ∇ u ϵ | 2 d x = 1 . In this note, we concern the compactness of the function family {uϵ}ϵ>0 and prove that up to a subsequence uϵ converges to some function u0 in C 1 (B ¯) as ϵ → 0. Furthermore, u0 is an extremal function of the supremum sup u ∈ W 0 1 , 2 (B) , ∫ B | ∇ u | 2 d x ≤ 1 ∫ B e 4 π u 2 d x. Let us explain the result in geometry. Denote ω 0 = d x 1 2 + d x 2 2 be the standard Euclidean metric. Define a conical metric ω ϵ = | x | 2 ϵ ω 0 for x ∈ B . Then the extremal family {uϵ}ϵ>0 of the following Trudinger-Moser functionals ∫ B e 4 π (1 + ϵ) u 2 d v w ϵ under the constraint u ∈ W 0 1 , 2 (B) and ∫ B | ∇ ω ϵ u | 2 d v ω ϵ ≤ 1 is compact as ϵ → 0. [ABSTRACT FROM AUTHOR]

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field. [ABSTRACT FROM AUTHOR]

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L 1 -optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp L p -Sobolev and L p -logarithmic Sobolev inequalities (both for p > 1 and p = 1 ) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities. [ABSTRACT FROM AUTHOR]

Our main result is a weighted fractional Poincaré–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré–Sobolev estimate with A p weights of Fabes–Kenig–Serapioni by means of a fractional type result in the spirit of Bourgain–Brezis–Mironescu. Examples are given to show that the corresponding L p -versions of weighted Poincaré inequalities do not hold for p > 1 . [ABSTRACT FROM AUTHOR]

In this research, we analyze the existence and multiplicity of nonnegative solutions for a class of non-local elliptic system with Dirichlet boundary conditions. The problem's nonlinearity does not typically satisfy the Ambrosetti–Rabinowitz condition and includes various non-linear terms characterized by concave-convex variables and variable exponent functions. We use minimization techniques and the properties of a monotone operator of type (S) + to establish the existence of positive solutions for the non-local elliptic system within fractional Musielak spaces. Also, we show an embedding result from fractional Musielak spaces to the weighted Musielak spaces. Our main results generalize some recent findings in the literature to non-smooth cases. [ABSTRACT FROM AUTHOR]

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed. [ABSTRACT FROM AUTHOR]

Let B be a unit ball in ℝ2, W 0 1 , 2 (B) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely, sup u ∈ W 0 1 , 2 (B) , ∫ B | ∇ u | 2 d x ≤ 1 ∫ B | x | 2 ϵ e 4 π (1 + ϵ) u 2 d x can be attained by some radially symmetric function u ϵ ∈ W 0 1 , 2 (B) with ∫ B | ∇ u ϵ | 2 d x = 1 . In this note, we concern the compactness of the function family {uϵ}ϵ>0 and prove that up to a subsequence uϵ converges to some function u0 in C 1 (B ¯) as ϵ → 0. Furthermore, u0 is an extremal function of the supremum sup u ∈ W 0 1 , 2 (B) , ∫ B | ∇ u | 2 d x ≤ 1 ∫ B e 4 π u 2 d x. Let us explain the result in geometry. Denote ω 0 = d x 1 2 + d x 2 2 be the standard Euclidean metric. Define a conical metric ω ϵ = | x | 2 ϵ ω 0 for x ∈ B . Then the extremal family {uϵ}ϵ>0 of the following Trudinger-Moser functionals ∫ B e 4 π (1 + ϵ) u 2 d v w ϵ under the constraint u ∈ W 0 1 , 2 (B) and ∫ B | ∇ ω ϵ u | 2 d v ω ϵ ≤ 1 is compact as ϵ → 0. [ABSTRACT FROM AUTHOR]

We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles. [ABSTRACT FROM AUTHOR]