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In this paper, we deal with the L p boundedness of rough Fourier integral operators T a , φ with amplitude a ⁢ (x , ξ) ∈ L ∞ ⁢ S ρ m and phase function φ ⁢ (x , ξ) ∈ L ∞ ⁢ Φ 2 which satisfies a measure condition. We show that T a , φ is bounded on L p for 1 ≤ p ≤ ∞ if m < n ⁢ (ρ − 1) p − ρ ⁢ (n − 1) 2 ⁢ p when 1 ≤ p ≤ 2 or m < n ⁢ (ρ − 1) 2 − ρ ⁢ (n − 1) 2 ⁢ (1 − 1 p) when 2 ≤ p ≤ ∞ . Our main results extend and improve some known results about L p boundedness of Fourier integral operators. [ABSTRACT FROM AUTHOR]

In this short note, we give an easy proof of the following result: for n ≥ 2 , lim t → 0 e i t Δ f x + γ (t) = f (x) almost everywhere whenever γ is an α -Hölder curve with 1 2 ≤ α ≤ 1 and f ∈ H s (R n) , with s > n 2 (n + 1) . This is the optimal range of regularity up to the endpoint. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . Assume that L R is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on L 2 (Ω) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of L R , the authors establish a new atomic characterization of the Hardy space H L R p (Ω) associated with L R . Using this, the authors further show that, for any given p ∈ (n n + δ 0 , 1 ] , H z p (Ω) + L ∞ (Ω) = H L N p (Ω) = H L R p (Ω) ⫋ H L D p (Ω) = H r p (Ω) , where H L D p (Ω) and H L N p (Ω) denote the Hardy spaces on Ω associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, H z p (Ω) and H r p (Ω) respectively denote the "supported type" and the "restricted type" Hardy spaces on Ω , and δ 0 ∈ (0 , 1 ] is the critical index depending on the operators L D , L N , and L R . The authors then obtain the boundedness of the Riesz transform ∇ L R - 1 / 2 on the Lebesgue space L p (Ω) when p ∈ (1 , ∞) [if p > 2 , some extra assumptions are needed] and its boundedness from H L R p (Ω) to L p (Ω) when p ∈ (0 , 1 ] or to H r p (Ω) when p ∈ (n n + 1 , 1 ] . As applications, the authors further obtain the global regularity estimates, in L p (Ω) when p ∈ (0 , p 0) and in H r p (Ω) when p ∈ (n n + 1 , 1 ] , for the inhomogeneous Robin problem of L R on Ω , where p 0 ∈ (2 , ∞) is a constant depending only on n, Ω , and the operator L R . The main novelties of these results are that the range (0 , p 0) of p for the global regularity estimates in the scale of L p (Ω) is sharp and that, in some sense, the space X : = H L R 1 (Ω) is also optimal to guarantee both the boundedness of ∇ L R - 1 / 2 from X to L 1 (Ω) or to H r 1 (Ω) and the global regularity estimate ‖ ∇ u ‖ L n n - 1 (Ω ; R n) ≤ C ‖ f ‖ X for inhomogeneous Robin problems with C being a positive constant independent of both u and f. [ABSTRACT FROM AUTHOR]

In this article we study the solution u(t, x) of the Cauchy problem of linear damped wave equation. We obtain the sufficient and necessary condition of the boundedness of the solution u(t, x) on the Triebel–Lizorkin space at a fixed time t. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R n + 1 , n ≥ 2 , be an open set with Ahlfors–David regular boundary satisfying the corkscrew condition. When Ω is quantitatively connected (i.e., it satisfies the Harnack chain condition) it is known that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class A ∞ ) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting, it is also known that these properties are stable under Fefferman–Kenig–Pipher's perturbations. Nonetheless, it has been an open problem to show whether, without connectivity, the previous two conditions remain equivalent or whether there is a Fefferman–Kenig–Pipher perturbation theory. In this paper we settle the question of whether, for general real elliptic operators in sets without any connectivity assumption, quantitative absolute continuity of the elliptic measure (expressed via a corona decomposition for the elliptic measure) and Carleson measure type estimates for bounded null solutions are equivalent. We show that any of these conditions is also equivalent to the fact that the Green function is comparable, in the corona sense, to the distance to the boundary. Our characterization has profound consequences. First, we extend Fefferman–Kenig–Pipher's perturbation result to sets which are not necessarily connected, and this is the first result in this general setting. Second, in the case of the Laplacian, and more generally for L 1 -Kenig–Pipher non-symmetric operators with variable coefficients, our conditions characterize the uniform rectifiability of the boundary. Last, our results generalize previous work in settings where quantitative connectivity is assumed. [ABSTRACT FROM AUTHOR]

In this paper, we are going to establish a reversed Hardy–Littlewood–Sobolev inequality on different dimensional space and prove the existence of extremal functions for the best constant. Furthermore, we investigate the regularity of extremal functions and provide a necessary conditions for the existence of solutions of the integral systems. Finally, we classify the extremal functions. [ABSTRACT FROM AUTHOR]

In dimensions d ∈ { 3 , 4 , 5 , 6 , 7 } , we prove that the constant functions on the unit sphere S d - 1 ⊂ R d maximize the weighted adjoint Fourier restriction inequality ∫ R d | f σ ^ (x) | 4 (1 + g (x)) d x 1 / 4 ⩽ C ‖ f ‖ L 2 (S d - 1) , where σ is the surface measure on S d - 1 , for a suitable class of bounded perturbations g : R d → C . In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ( g = 0 ), this was established by Foschi ( d = 3 ) and by the first and third authors ( d ∈ { 4 , 5 , 6 , 7 } ) in 2015. Our methods also yield a new sharp adjoint restriction inequality on S 7 ⊂ R 8 . [ABSTRACT FROM AUTHOR]

Motivated by the recent works [1, 23], we study the following extension of Calderón–Zygmund type singular integrals T β f (x) = p. v. ∫ R n Ω (y) | y | n - β f (x - y) d y , for 0 < β < n , and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt A p -weighted L p -spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small β , and therefore one can pass onto the limits as β → 0 to deduce analogous estimates for the classical Calderón–Zygmund type singular integrals and their commutators. [ABSTRACT FROM AUTHOR]

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups. [ABSTRACT FROM AUTHOR]

We prove sharp weighted estimates for the non-tangential maximal function of singular integrals mapping functions from R n to the half-space in R 1 + n above R n . The proof is based on pointwise sparse domination of the adjoint singular integrals that map functions from the half-space back to the boundary. It is proved that these map L 1 functions in the half-space to weak L 1 functions on the boundary. From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages. [ABSTRACT FROM AUTHOR]

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities. [ABSTRACT FROM AUTHOR]

We prove the existence of functions that extremize the endpoint L 2 to L 4 adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space R 4 and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer. [ABSTRACT FROM AUTHOR]

It is known that the damped fractional wave equation has the diffusive structure as t → ∞ . Let u (t , x) = e − t cosh (t L) f (x) + e − t sinh (t L) L (f (x) + g (x)) be the solution of the Cauchy problem for the damped fractional wave equation, where L involves the fractional Laplacian (− △) α on the space variable. We can study the decay estimate of the solution u (t , x) over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0 < α < 1 , the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α = 1 , the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate. [ABSTRACT FROM AUTHOR]

In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ which are integrable on β-dimensional subsets of the cube Q 0 : ∥ u ∥ BMO β ⁢ (Q 0) := sup Q ⊂ Q 0 ⁡ inf c ∈ ℝ ⁡ 1 l ⁢ (Q) β ⁢ ∫ Q | u - c | ⁢ 푑 ℋ ∞ β , where the supremum is taken over all finite subcubes Q parallel to Q 0 , l ⁢ (Q) is the length of the side of the cube Q, and ℋ ∞ β is the Hausdorff content. In the case β = d we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ (0 , d ] one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C > 0 such that ℋ ∞ β ⁢ ({ x ∈ Q : | u ⁢ (x) - c Q | > t }) ≤ C ⁢ l ⁢ (Q) β ⁢ exp ⁡ (- c ⁢ t ∥ u ∥ BMO β ⁢ (Q 0) ) for every t > 0 , u ∈ BMO β ⁢ (Q 0) , Q ⊂ Q 0 , and suitable c Q ∈ ℝ . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest. [ABSTRACT FROM AUTHOR]

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that ∇ G (X) G (X) - ∇ D (X) D (X) 2 D (X) d X is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's α -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above. [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we prove bilinear sparse domination bounds for a wide class of Fourier integral operators of general rank, as well as oscillatory integral operators associated to Hörmander symbol classes S ρ , δ m for all 0 ≤ ρ ≤ 1 and 0 ≤ δ < 1 , a notable example is the Schrödinger operator. As a consequence, one obtains weak (1, 1) estimates, vector-valued estimates, and a wide range of weighted norm inequalities for these classes of operators. [ABSTRACT FROM AUTHOR]

This paper is an exposition of some estimates which have a number of applications to interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable estimates. In Abilov et al. (Comput Math Math Phys 48:2146, 2008) , Abilov et al. proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilov for only two variables, using a translation operator. The purpose of this paper is to study these estimates for measurable sets from complex domain to hyper complex domain by using quaternion algebras, associated with the quaternion linear canonical transform, constructed by the generalized Steklov function. [ABSTRACT FROM AUTHOR]

In this article, we studied the convolution operators M k with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator M k is associated to the characteristic hypersurfaces Σ ⊂ R 3 of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order - k for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point v ∈ Σ at which, exactly one of the principal curvatures of the surface Σ does not vanish. Such surfaces exhibit singularities of the type A in the sense of Arnold’s classification. Denoting by k p the minimal number such that M k is L p ↦ L p ′ -bounded for k > k p , we showed that the number k p depends on some discrete characteristics of the surface Σ . [ABSTRACT FROM AUTHOR]

We consider the solutions of the Klein–Gordon equation in the 2 + 1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone. [ABSTRACT FROM AUTHOR]

Getting inspired by the Caffarelli–Silvestre extensions, this paper investigates the weighted Lorentz–Sobolev capacities and their capacitary strong inequalities with applications to the Sobolev-type embeddings. Consequently, the weighted Lebesgue-Sobolev capacities and their applications to a functional inequality problem and the existence-regularity of solutions to the prototype p-Laplace equations with weight are addressed as well. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 푛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 . [ABSTRACT FROM AUTHOR]

Let (λ k) be a strictly increasing sequence of positive numbers such that ∑ k = 1 ∞ 1 λ k < ∞ . Let f be a bounded smooth function and denote by u = u f the bounded classical solution to u (x) - 1 2 ∑ k = 1 m D kk 2 u (x) + ∑ k = 1 m λ k x k D k u (x) = f (x) , x ∈ R m. It is known that the following dimension-free estimate holds: ∫ R m ∑ k = 1 m λ k (D k u (y)) 2 p / 2 μ m (d y) ≤ (c p) p ∫ R m | f (y) | p μ m (d y) , 1 < p < ∞ where μ m is the "diagonal" Gaussian measure determined by λ 1 , ... , λ m and c p > 0 is independent of f and m. This is a consequence of generalized Meyer's inequalities [4]. We show that, if λ k ∼ k 2 , then such estimate does not hold when p = ∞ . Indeed we prove sup f ∈ C b 2 (R m) , ‖ f ‖ ∞ ≤ 1 ∑ k = 1 m λ k (D k u f (0)) 2 → ∞ as m → ∞. This is in contrast to the case of λ k = λ > 0 , k ≥ 1 , where a dimension-free bound holds for p = ∞ . [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with L p boundary data, 1 ≤ p ≤ ∞ . We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc. [ABSTRACT FROM AUTHOR]

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case—that is, when the Hilbert space is the regular representation of the Fuchsian group, Γ . Second, when Γ ⊂ PSU (1 , 1) acts isometrically on the hyperbolic plane H and the Hilbert space is L 2 H , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over Σ = H / Γ and transforms the hyperbolic Laplacian into the covariant one. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

This article focuses on the Cauchy problem of the Benjamin–Bona–Mahony–Burgers (BBMB) equation with nonlocal control constraints in ℝ 3 . We employ the Phragmén–Lindelöf decomposition method and Green's function to investigate global classical solutions and their long-term behaviors, including optimal estimates for finite density initial perturbation. Additionally, optimal solutions near the minimal mass ground state are analyzed. [ABSTRACT FROM AUTHOR]

A polynomial-in-time growth bound is established for global Sobolev H s (T) solutions to the derivative nonlinear Schrödinger equation on the circle with s > 1 . These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution. [ABSTRACT FROM AUTHOR]

In this article, we study the Schrödinger equation associated with the Weinstein operator (SW). We prove that the problem (SW) is locally well-posed in L α 2 R + d + 1 of the subcritical case p ∈ 0 , 4 d + 2 α + 2 and we show that the equation (SW) is globally well-posed in the critical case p = 4 d + 2 α + 2 . [ABSTRACT FROM AUTHOR]

Consider a pseudo-differential operator where the symbol a is in the rough Hörmander class with m ∈ ℝ and ρ ∈ [0, 1]. In this note, when 1 ≤ p ≤ 2, if m < n (ρ − 1) p and , then for any f ∈ S(ℝn) and x ∈ ℝn, we prove that where M is the Hardy-Littlewood maximal operator. Our theorem improves the known results and the bound on m is sharp, in the sense that n (ρ − 1) p can not be replaced by a larger constant. [ABSTRACT FROM AUTHOR]

Let λ ∈ (- 1 2 , ∞) , and { W t λ } t > 0 be the heat semigroup related to the Bessel Schrödinger operator S λ : = - d 2 d x 2 + λ 2 - λ x 2 on R + : = (0 , ∞) . The authors introduce the weighted Morrey–Campanato space BMO α (R + , ω) with α ∈ [ 0 , 1) and ω ∈ A ∞ (R +) , and show that for any weight function ω ∈ R H s ′ (R +) ∩ A p / s (R +) , the oscillation, variation, radial maximal operator, and maximal operator of difference associated with the family { t m ∂ t m W t λ } t > 0 are bounded from BMO α (R + , ω) to its subspace BLO α (R + , ω) , where λ ∈ R + , m ∈ N ∪ { 0 } , p ∈ (1 , ∞) , s ∈ [ 1 , p) such that p / s + α < 1 + min { 1 , λ } , and s ′ denotes the conjugate exponent of s. These results are new even in the case of ω ≡ 1 . As a corollary, the boundedness of these operators on spaces BMO α (R + , ω) is further established. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of extremals for the following singular critical Trudinger–Moser inequality with logarithmic weights: sup u ∈ W 0 , r 1 , n (B , ω β) , u ω β ≤ 1 ∫ B exp (α n , β , σ u n n - 1 1 - β ) x σ d x < ∞ , where B is the unit ball in R n , β ∈ [ 0 , 1) , σ ∈ [ 0 , n) , α n , β , σ = n - σ [ ω n - 1 1 n - 1 1 - β ] 1 1 - β , W 0 , r 1 , n (B , ω β) denotes the radial weighted Sobolev space with the norm u ω β = ∫ B ∇ u n ω β x d x 1 n , ω β (x) = (log e x ) β (n - 1) . Moreover, for m > 0 , we establish a singular supercritical Trudinger–Moser inequality with logarithmic weights sup u ∈ W 0 , r 1 , n (B , ω β) , u ω β ≤ 1 ∫ B exp ((α n , β , σ + x m) u n n - 1 1 - β ) x σ d x < ∞ , and prove the existence of its extremal functions. [ABSTRACT FROM AUTHOR]

We prove that for any second-order, homogeneous, N × N elliptic system L with constant complex coefficients in R n , the Dirichlet problem in R + n with boundary data in CMO (R n - 1 , C N) is well-posed under the assumption that d μ (x ′ , t) : = | ∇ u (x) | 2 t d x ′ d t is a strong vanishing Carleson measure in R + n in some sense. This solves an open question posed by Martell et al. [1, Open Question 8], which wishes to formulate and prove a well-posedness result for the Dirichlet problem in R + n for L as above, and establish a Fatou-type theorem for null-solutions of L. Our proof relies on a quantitative Fatou-type theorem, which not only guarantees the existence of the pointwise nontangential boundary trace for smooth null-solutions satisfying a strong vanishing Carleson measure condition, but also includes a Poisson integral representation formula of solutions along with a characterization of CMO (R n - 1 , C N) in terms of the traces of solutions of elliptic systems. Moreover, we are able to establish the well-posedness of the Dirichlet problem in R + n for a system L as above in the case when the boundary data belongs to XMO (R n - 1 , C N) , which lies in between CMO (R n - 1 , C N) and VMO (R n - 1 , C N) . Analogously, we formulate a new brand of vanishing Carleson measure conditions and a characterization of XMO (R n - 1 , C N) in terms of the traces of solutions of elliptic systems. [ABSTRACT FROM AUTHOR]

We consider general cooperative elliptic systems possibly without variational structure and with differential operator resembling that from an Euler–Lagrange equation for a sharp Hardy–Sobolev inequality. Under suitable growth conditions on the source nonlinearities and geometric assumptions on the domain, we derive various existence and non-existence results and Liouville theorems. The results are obtained by incorporating and adapting various techniques, including variants of the method of moving planes enhanced by Kelvin and Emden–Fowler type transformations, as well as degree theoretic shooting methods. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for interpolating a sequence by a function belonging to the classical weighted harmonic Bergman space defined on the unit real ball. [ABSTRACT FROM AUTHOR]

We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John–Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John–Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John–Nirenberg inequality. Moreover, we consider John–Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition. [ABSTRACT FROM AUTHOR]

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator and other compensated compactness quantities map their natural domain of definition onto the real-variable Hardy space H 1 (R n) . We present an axiomatic, Banach space geometric approach to the problem in the case of quadratic operators. We also make progress on the main open case, the Jacobian equation in the plane. [ABSTRACT FROM AUTHOR]

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C ⁢ (Y) = (0 , ∞) r × Y equipped with the metric g = d ⁢ r 2 + r 2 ⁢ h , where the cross section Y is a compact (n - 1) -dimensional closed Riemannian manifold (Y , h) . Assuming the initial data possesses additional regularity in the angular variable θ ∈ Y , we prove some linear restriction estimates for the solutions of Schrödinger equations on the cone X. The smallest positive eigenvalue of the operator Δ h + V 0 + (n - 2) 2 / 4 plays an important role in the result. As applications, we prove local energy estimates and Keel–Smith–Sogge estimates for the Schrödinger equation in this setting. [ABSTRACT FROM AUTHOR]