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Let A be a general expansive matrix. Let X be a ball quasi-Banach function space on R n , which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first establish the boundedness of convolutional anisotropic Calderón–Zygmund operators on the Hardy space H X A (R n) . As an application, the authors also obtain the boundedness of Fourier multipliers satisfying anisotropic Mihlin conditions on H X A (R n) . All these results have a wide range of applications; in particular, when they are applied to Lebesgue spaces, all these results reduce back to the known best results and, even when they are applied to Lorentz spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces, the obtained results are also new. [ABSTRACT FROM AUTHOR]

We obtain polynomial decay rates for C 0 -semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups, we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces. [ABSTRACT FROM AUTHOR]

Let p> 2 be a given exponent. In this paper, we prove, with the best constant, the weak-type (p,p) inequality ∥ T m f ∥ L p , ∞ (R d) ⩽ C p ∥ f ∥ L p (R d) for a large class of non-symmetric Fourier multipliers Tm obtained via modulation of jumps of certain Lévy processes. In particular, the estimate holds for appropriate linear combinations of second-order Riesz transforms and skew versions of the Beurling-Ahlfors operator on the complex plane. The proof rests on a novel probabilistic bound for Hilbert-space-valued martingales satisfying a certain non-symmetric subordination principle. Further applications to harmonic functions and Riesz systems on Euclidean domains are indicated. [ABSTRACT FROM AUTHOR]

Let Φ i , Ψ i be Young functions, ω i be weights and M ω i Φ i , Ψ i (R d) be the corresponding Orlicz modulation spaces for i = 1 , 2 , 3 . We consider linear (respect. bilinear) multipliers on R d , that is bounded measurable functions m (ξ) (respect. m (ξ , η) ) on R d (respect. R 2 d ) such that T m (f) (x) = ∫ R d f ^ (ξ) m (ξ) e 2 π i ⟨ ξ , x ⟩ d ξ (respect. B m (f 1 , f 2) (x) = ∫ R d ∫ R d f 1 ^ (ξ) f 2 ^ (η) m (ξ , η) e 2 π i ⟨ ξ + η , x ⟩ d ξ d η define a bounded linear (respect. bilinear) operator from M ω 1 Φ 1 , Ψ 1 (R d) to M ω 2 Φ 2 , Ψ 2 (R d) (respect. M ω 1 Φ 1 , Ψ 1 (R d) × M ω 2 Φ 2 , Ψ 2 (R d) to M ω 3 Φ 3 , Ψ 3 (R d) ). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces. [ABSTRACT FROM AUTHOR]

In the present paper, using the theory of boundary values of analytic semigroups, we find necessary and sufficient conditions to guarantee that the operator i (- Δ) α / 2 generates a strongly continuous semigroup in L p (R n) . [ABSTRACT FROM AUTHOR]

For an integer d ≥ 2 , t ∈ R , and a 0-homogeneous function Φ ∈ C ∞ (R d \ { 0 } , R) , we consider the family of Fourier multiplier operators T Φ t associated with symbols ξ ↦ exp (i t Φ (ξ)) and prove that for a generic phase function Φ , one has the estimate ‖ T Φ t ‖ L p → L p ≳ d , p , Φ ⟨ t ⟩ d | 1 p - 1 2 | . That is the maximal possible order of growth in t → ± ∞ , according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions Φ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon. [ABSTRACT FROM AUTHOR]

We study rates of decay for C 0 -semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with | s | β log (| s |) b , β , b ≥ 0 , as | s | → ∞ , and with | s | - α log (1 / | s |) a , α , a ≥ 0 , as | s | → 0 . Our results do not suppose that the semigroup is bounded. In particular, for a = b = 0 , our results improve the rates involving Fourier types obtained by Rozendaal and Veraar (J Funct Anal 275(10):2845–2894, 2018). [ABSTRACT FROM AUTHOR]

Let ... and t, r, p ∈ (0, ∞) . In this paper, we introduce the slice Besov-type space ... and the slice Triebel-Lizorkin-type space ..., and establish their φ-transform characterizations in the sense of Frazier and Jawerth. The embedding properties, characterizations via the Peetre maximal function, the Lusin area function, smooth atomic and molecular decompositions of these spaces are also obtained. As applications, we obtain the boundedness on these spaces of Fourier multipliers with symbols satisfying some generalized Hörmander condition. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the degenerate fractional differential equations with finite delay: Dα(Mu)(t)+cDβ(Mu)(t)$D^\alpha (Mu)(t) + cD^\beta (Mu)(t)$=Au(t)+Fut+f(t),(0≤t≤2π)$= Au(t) + Fu_t + f(t),(0\le t\le 2\pi)$ on Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, where A$A$ and M$M$ are closed linear operators in a complex Banach space X$X$ satisfying D(A)⊂D(M)$D(A)\subset D(M)$, c∈C$c\in \mathbb {C}$ and 0<β<α$0 < \beta < \alpha$ are fixed, ut(s)=u(t+s)$u_t(s) = u(t+s)$ when 0≤t≤2π,−2π≤s≤0$0\le t\le 2\pi , -2\pi \le s\le 0$, and the delay operator F$F$ is a bounded linear operator from Lp([−2π,0];X)$L^p([-2\pi ,0]; X)$ (resp. Bp,qs([−2π,0];X)$B_{p,q}^s([-2\pi ,0]; X)$) into X$X$. Using known operator‐valued Fourier multiplier theorems on Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, we completely characterize the Lp$L^p$‐well‐posedness and the Bp,qs$B_{p,q}^s$‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems. [ABSTRACT FROM AUTHOR]

We investigate the Gabor spherical mean transform and demonstrate the quantitative Shapiro dispersion uncertainty principle and the umbrella theorem specifically tailored for this transform. Next, we define the Toeplitz operators L S g 1 , g 2 in connection with two window functions g 1 and g 2 , as well as the symbol S . We establish the boundedness and compactness of these operators. Ultimately, we introduce the Schatten–von Neumann class S p , where p ∈ [ 1 , + ∞ ] , and show that the Toeplitz operators are members of the class S p . Additionally, we provide a proof for a trace formula. [ABSTRACT FROM AUTHOR]

Let G be a locally compact quantum group with dual G ^ . Suppose that the left Haar weight φ and the dual left Haar weight φ ^ are tracial, e.g. G is a unimodular Kac algebra. We prove that for 1 < p ≤ 2 ≤ q < ∞ , the Fourier multiplier m x is bounded from L p (G ^ , φ ^) to L q (G ^ , φ ^) whenever the symbol x lies in L r , ∞ (G , φ) , where 1 / r = 1 / p - 1 / q . Moreover, we have ‖ m x : L p (G ^ , φ ^) → L q (G ^ , φ ^) ‖ ≤ c p , q ‖ x ‖ L r , ∞ (G , φ) , where c p , q is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for R n , and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of L p -Fourier multipliers over discrete group von Neumann algebras. A similar result for S p - S q Schur multipliers is also proved. [ABSTRACT FROM AUTHOR]

We mainly study the dimension-free Lp{L}^{p}-inequality of the truncated maximal operator Mnaf(x)=supt>01∣Ba1∣∫Ba1f(x−ty)dy,{M}_{n}^{a}f\left(x)=\mathop{\sup }\limits_{t\gt 0}\frac{1}{| {B}_{a}^{1}| }\left|\mathop{\int }\limits_{{B}_{a}^{1}}f\left(x-ty){\rm{d}}y\hspace{-0.25em}\right|, where Ba1={x:a≤∣x∣≤1}{B}_{a}^{1}=\left\{x:a\le | x| \le 1\right\}. When 0≤an/(n−1)p\gt n\hspace{0.1em}\text{/}\hspace{0.1em}\left(n-1). When a=1a=1, we prove that ‖Mn1‖Lp(Rn)≤C(p)‖f‖Lp(Rn)\Vert {M}_{n}^{1}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})} for p≥2p\ge 2.

We study Fourier multiplier operators associated with symbols \xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |)), where \lambda is a real number and \phi is a real-valued \mathrm {C}^\infty function on the standard unit sphere \mathbb {S}^{n-1}\subset \mathbb {R}^n. For 1

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′(t)+Λu′(t)+∫−∞ta(t−s)u′(s)ds=Au(t)+∫−∞tb(t−s)Bu(s)ds+f(t)(P)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$with infinite delay on [0, 2π] in Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s (\mathbb {T}; X)$, where A,B,Λ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(Λ)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda)$ and the kernels a,b∈L1(R+)$ a, b\in L^1(\mathbb {R}_+)$. Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs$B_{p,q}^s$(T;X)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of linear operators defined on a separable Hilbert space H . More precisely, we define the generalized Fourier multipliers T σ A : H → H , where σ are symbols which belong to L p (Z) , 1 ≤ p ≤ ∞ and A : H → H is a bounded linear operator which satisfies some suitable conditions. Moreover, we also define function of generalized Fourier multipliers F T σ A : H → H where F : C → C is a continuous function. We give some boundedness and compactness results and we also study Schatten-von Neumann properties for these classes of linear operators mentioned above. Further, we study the existence and the regularity of the solutions of the linear equations F T σ A f + λ f = g , f , g ∈ H , where F : R → R + is a real-valued non-negative continuous function and λ is a real positive number. At last, an example of function of generalized Fourier multipliers in connection to Weyl transform is given. [ABSTRACT FROM AUTHOR]

We give an overview of some recent results on operator-valued (L p , L q) Fourier multipliers and stability theory for evolution equations. The aim is to provide a relatively nontechnical introduction to the underlying ideas, emphasizing the connection between the two areas. We also indicate how operator-valued (L p , L q) Fourier multipliers can be applied to functional calculus theory. [ABSTRACT FROM AUTHOR]

Let p (·) : R → (1 , ∞) be a sufficiently regular variable exponent and ϱ be a Khvedelidze weight on R . Suppose that a function a belongs to the algebra P C p (·) , ϱ of piecewise continuous Fourier multipliers on the weighted variable Lebesgue space L p (·) (R , ϱ) . We show that the Fourier convolution operator W 0 (a) = F - 1 a F is invertible on the space L p (·) (R , ϱ) if and only if its symbol a is invertible in L ∞ (R) . [ABSTRACT FROM AUTHOR]

In this paper, we consider the global regularity of 2D magnetic Bénard fluid equations with almost magnetic diffusion and thermal diffusivity and without kinematic viscosity. We focus on this goal in two ways. In one way, the magnetic diffusion and thermal diffusivity are separately given by D 2 and D 3 two Fourier multipliers whose symbols m 2 and m 3 are respectively given by m 2 (ξ) ≡ | ξ | 2 log ⁡ (e + | ξ | 2) β and m 3 (ξ) ≡ | ξ | 2 log ⁡ (e + | ξ | 2) γ . In another way, we generalize the previous case and the magnetic diffusion and thermal diffusivity are separately given by L 2 and L 3 two Fourier multipliers whose symbols m 2 and m 3 are respectively given by m 2 = | ξ | 2 g 2 (ξ) and m 3 = | ξ | 2 g 3 (ξ) , where g j = g j (| ξ |) (j = 2 , 3) are two radial non-decreasing smooth functions. [ABSTRACT FROM AUTHOR]

Asmar et al. [Note on norm convergence in the space of weak type multipliers. J Operator Theory. 1998;39(1):139–149] proved that the space of weak-type Fourier multipliers acting from L p into L p , ∞ is continuously embedded into L ∞ . We obtain a sharper result in the setting of abstract Lorentz spaces Λ q (X) with 0 < q ≤ ∞ built upon a Banach function space X on R n . We consider a source space S and a target space T in the class of admissible spaces A := { X , Λ q (X) : 0 < q ≤ ∞ }. Let M S , T 0 denote the space of Fourier multipliers acting from S to T . We show that if the space X satisfies the weak doubling property, then the space M Λ q (X) , Λ ∞ (X) 0 is continuously embedded into L ∞ for every 0 < q ≤ ∞. This implies that M S , T 0 is a quasi-Banach space for all choices of source and target spaces S , T ∈ A . [ABSTRACT FROM AUTHOR]

We refine the decay estimate of the heat semigroup {T(t)}t≥0defined on homogeneous Besov spaces Ḃp,qs(Rn)for s∈R,p,q∈[1,∞], which is obtained by Kozono et al. (2003). In particular, we give an explicit representation of a constant appeared in the decay estimate of {T(t)}t≥0, which provides a space–time analytic smoothing effect of {T(t)}t≥0. As a by-product, we obtain a radius of convergence of the Taylor expansion exactly. Furthermore, it is also showed that {T(t)}t≥0is a bounded analytic C0-semigroup on Ḃp,qs(Rn)for s∈R,p,q∈[1,∞), where {T(t)}t≥0can be extended as an analytic function of t on the sector {t∈ℂ∖{0}||argt|

We establish the strong L p -inequality for the maximal function over hyper-surface based on Euclidean ball sup t > 0 1 | B n | ∫ B n f (x - t u , y - t 2 | u | 2) d u . We prove the L p -boundedness of the maximal function for p > 1 . Specially, when p = 2 , we prove the L 2 -boundedness of the maximal function can be controlled by C n 7 8 , where C is independent of n. [ABSTRACT FROM AUTHOR]

Let 1 < p < q ≤ 2. The paper contains the identification of the best constant Cp,q such that the following holds. If X, Y are Hilbert-space valued martingales such that Y is differentially subordinate to X, then we have ||Y||p,∞ ≤ Cp,q ||X||p,q The proof rests on the careful combination of Burkholder's method and optimization arguments. As an application, related sharp Lorentz-norm inequalities for a wide class of Fourier multipliers are obtained. [ABSTRACT FROM AUTHOR]

We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from C c l (X) to ( C c m (Y)) ′ by kernels, which we prove to be distributions of order l in x , but higher, although still finite, order in y. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in x ∈ R d , and of finite order in ξ ∈ S d − 1 . This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations. [ABSTRACT FROM AUTHOR]

We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance. [ABSTRACT FROM AUTHOR]

In this paper, we give an equivalent characterization of Triebel-Lizorkin spaces. This reveals the equivalent relation between the mixed derivative norm and single variable norm. Complex interpolation and Littlewood-Paley decomposition are applied in our proofs. [ABSTRACT FROM AUTHOR]

We extend results on the invertibility of Fourier convolution operators with piecewise continuous symbols on the Lebesgue space Lp(R), p E (1, oe), obtained by Roland Duduchava in the late 1970s, to the setting of a separable Banach function space X(R) such that the Hardy--Littlewood maximal operator is bounded on X(R) and on its associate space XZ(R). We specify our results in the case of rearrangement-invariant spaces with suitable Muckenhoupt weights. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the fractional degenerate integro‐differential equations (Pα):Dα(Mu)(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(t∈T:=[0,2π]),in Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A, B and M are closed linear operators on a Banach space X satisfying D(A)∩D(B)⊂D(M), D(A)∩D(B)≠{0}, α>0 and a,b∈L1(R+). We completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using operator‐valued Fourier multiplier. We also give an example that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

We study membership in Lipschitz classes Λ β (D) for a class of α-harmonic functions in the open unit disc D in the complex plane. From earlier work by Olofsson and Wittsten we know that such an α-harmonic function u is the α-harmonic Poisson integral P α [ f ] of its boundary value function f on the unit circle T . We determine when the Poisson integral P α [ f ] belongs to a Lipschitz class Λ β (D) for the unit disc. [ABSTRACT FROM AUTHOR]

The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal u = 1 / w is a (p, q)-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift-invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian–Low uncertainty principle in these settings. [ABSTRACT FROM AUTHOR]

We establish Littlewood–Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate bi-parameter modulation spaces on the product of two Euclidean spaces Rn and Rm via uniform decompositions of each factor. A molecular decomposition of these bi-parameter spaces are given, which generalizes the related single-parameter result of Kobayashi and Sawano [33]. Furthermore, we prove the boundedness of a class of Fourier multipliers on bi-parameter modulation spaces, generalizing the results of Bényi et al. [2] and Feichtinger and Narimani [17]. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the fractional differential equations with infinite delay (Pα):Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(0≤t≤2π),on Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A)∩D(B)≠{0},  α>0 and a,b∈L1(R+). Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied. [ABSTRACT FROM AUTHOR]