Space Mp Research Articles

Two-weighted inequalities for maximal operator in generalized weighted Morrey spaces on spaces of homogeneous type

In this paper we give a characterization of two-weighted inequalities for maximal operators in generalized weighted Morrey spaces on spaces of homogeneous type Mp,φω (X). We prove the boundedness of maximal operator M from the spaces Mp,φ1ωδ 1(X)to the spaces Mp,φ2ωδ2(X), where 1 < p < ∞, 0 < δ < 1 and (ω1, ω2) ∈ Aep(X).

A measure characterization of embedding and extension domains for Sobolev, Triebel–Lizorkin, and Besov spaces on spaces of homogeneous type

In this article, for an optimal range of the smoothness parameter s that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajłasz–Triebel–Lizorkin spaces Mp,qs and Hajłasz–Besov spaces Np,qs in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results of this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces.

On the convolution operator in Morrey spaces

This paper is devoted to the study of upper bounds for the norm of the convolution operator in Morrey spaces. The spaces Mp,qα, which cover the classical Morrey spaces, are introduced. Moreover, their embedding properties are investigated, and their interpolation properties are described. Young-O'Neil type inequalities in Morrey spaces are proved. New results on the boundedness of Riesz's potential in Morrey spaces are established.

Nonhomogeneous central Morrey-type spaces in Lp(⋅) and weak estimates for the maximal and Riesz potential operators

Our aim in this paper is to discuss the weak estimate for the maximal and potential operators in the nonhomogeneous central Morrey-type space Mp(⋅),q,ν(Rn), which consists of all measurable functions f on Rn satisfying ‖f‖Mp(⋅),q,ν(Rn)=(∫1∞(r−ν‖f‖ Lp(⋅)(B(0,r)))qdr r)1∕q<∞, where ν>0, 0<q<∞ and p(⋅) is a variable exponent satisfying the log-Hölder condition at infinity. When q=∞, we apply a necessary modification.

The embedding property of the scaling limit of modulation spaces

Modulation spaces Mp,qs were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [1], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as Lp spaces, Fourier Lp spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.

On the existence of global solutions of the Hartree equation for initial data in the modulation space [formula omitted

In this paper, we study the Cauchy problem for Hartree type equationiut+uxx=[K⁎|u|2]u, with Cauchy data in modulation spaces Mp,q(R). We establish global well-posedness results in Mp,p′(R) when K(x)=λ|x|γ,(λ∈R,0<γ<1) with no smallness condition on initial data, where p′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces.

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator H=−Δ+|x|2 in Rd and its fractional powers Hβ, β>0 in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform Vgf (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of Vgf, that is in terms of membership to modulation spaces Mp,q, 0<p,q≤∞. We prove the complete range of fixed-time estimates for the semigroup e−tHβ when acting on Mp,q, for every 0<p,q≤∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing.As an application, we establish global well-posedness for the nonlinear heat equation for Hβ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e−ct as the solution of the corresponding linear equation, where c=dβ is the bottom of the spectrum of Hβ. Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to M∞,1).

The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

We study a natural question that, apparently, has not been well addressed in the literature. Given functions u with support in the unit ball B1⊂Rn and with gradient in the Morrey space Mp,λ(B1), where 1<p<λ<n, what is the largest range of exponents q for which necessarily u∈Lq(B1)? While David R. Adams proved in 1975 that this embedding holds for q≤λp/(λ−p), an article from 2011 claimed the embedding in the larger range q<np/(λ−p). Here we disprove this last statement by constructing a function that provides a counterexample for q>λp/(λ−p). The function is basically a negative power of the distance to a set of Hausdorff dimension n−λ. When λ∉Z, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent q can go up to np/(λ−p).

Time-decay and Strichartz estimates for the BBM equation on modulation spaces: Existence of local and global solutions

In this paper we derive time-decay and Strichartz estimates for the generalized Benjamin-Bona-Mahony equation on the framework of modulation spaces Mp,qs. We use these results to analyze the existence of local and global solutions of the corresponding Cauchy problem with rough data in modulation spaces. As consequence of the existence theorems in modulation spaces, some results in Sobolev spaces are derived.

Decay and smoothness for eigenfunctions of localization operators

We study decay and smoothness properties for eigenfunctions of compact localization operators Aaφ1,φ2. Operators Aaφ1,φ2 with symbols a in the wide modulation space Mp,∞ (containing the Lebesgue space Lp), p<∞, and windows φ1,φ2 in the Schwartz class S are known to be compact. We show that their L2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces Mvs⊗1∞(R2d), s>0 (subspaces of Mp,∞(R2d), p>2d/s) the L2-eigenfunctions of Aaφ1,φ2 are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.

Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity (i∂tu−(−Δ)α2u±(K⁎|u|2)u=0) with Cauchy data in the modulation spaces Mp,q(Rd). For K(x)=|x|−γ(0<γ<min{α,d/2}), we establish global well-posedness results in Mp,q(Rd)(1≤p≤2,1≤q<2d/(d+γ)) when α=2,d≥1, and with radial Cauchy data when d≥2,2d2d−1<α<2. Similar results are proven in Fourier algebra FL1(Rd)∩L2(Rd).

The Dirichlet problem in a class of generalized weighted Morrey spaces

We show continuity in generalized weighted Morrey spaces Mp,'(w) of sub-linear integral operators generated by some classical integral operators and commutators. The obtained esti- mates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators with dis- continuous data.

Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

In this paper, we study the boundedness of the fractional integral operator I α on Carnot group G in the generalized Morrey spaces M p, φ (G). We shall give a characterization for the strong and weak type boundedness of I α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.

Modulation spaces and non-linear Hartree type equations

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u)=(k⋆|u|2)u under a specified condition on potential k with Cauchy data in modulation spaces Mp,q(Rn). We establish global well-posedness results in Mp,p(Rn) with 1≤p<2nn+ν, when k(x)=λ|x|ν(λ∈R,0<ν<min{2,n2}); in Mp,q(Rn) with 1≤q≤p≤2, when k∈M∞,1(Rn).

On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space [formula omitted

We prove global existence for the one-dimensional cubic nonlinear Schrödinger equation in modulation spaces Mp,p′ for p sufficiently close to 2. In contrast to known results, [9] and [14], our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas–Vega, Hyakuna–Tsutsumi and Grünrock to the modulation space setting and exploits polynomial growth of the free Schrödinger group on modulation spaces.

Rademacher functions in Morrey spaces

The Rademacher sums are investigated in the Morrey spaces Mp,w on [0,1] for 1≤p<∞ and weight w being a quasi-concave function. They span l2 space in Mp,w if and only if the weight w is smaller than log2−1/2⁡2t on (0,1). Moreover, if 1<p<∞ the Rademacher subspace Rp,w is complemented in Mp,w if and only if it is isomorphic to l2. However, the Rademacher subspace R1,w is not complemented in M1,w for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces Mp,w is described. It turns out that for any infinite-dimensional subspace X of Rp,w the following alternative holds: either X is isomorphic to l2 or X contains a subspace which is isomorphic to c0 and is complemented in Rp,w.

Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

AbstractWe obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey spaceMp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey spaceW˙2,1p,φ(Q,ω)$\dot W_{2,1}^{p,\varphi }\left( {Q,\omega } \right)$.

The Cauchy problem for the Hartree type equation in modulation spaces

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u)=(K∗|u|2)u under a specified condition on potential K with Cauchy data in modulation spaces Mp,q(Rd). We establish global well-posedness results in M1,1(Rd) when K(x)=λ|x|−γ(λ∈R,0<γ<min{2,d/2}); in Mp,q(Rd)(1≤q≤min{p,p′} where p′ is the Hölder conjugate of p∈[1,2]) when K is in Fourier algebra FL1(Rd), and local well-posedness result in Mp,1(Rd)(1≤p≤∞) when K∈M1,∞(Rd).

Sampling measures for the Gabor transform

Sampling measures of a space H are measures μ such that ∫|f|pdμ≈‖f‖p for every function f∈H. Here we present a study of these measures for the specific case of model spaces of the Gabor transform. These spaces are the continuous transforms Vφg of functions g in a modulation space Mp,q with respect to a fixed window φ∈M1, the Feichtinger Algebra.We obtain a characterization of these measures in terms of discrete sampling sets and stability results. For a special class of windows that includes most of the important examples of Gabor atoms we also find sufficient conditions in terms of weak limits of uniqueness sets. We also discuss some applications.

Marcinkiewicz integrals associated with Schrödinger operator on generalized Morrey spaces

Let L = −Δ+V be aS chroperator, where Δ is the Laplacian on R n , while nonnegative potential V belongs to the reverse Holder class. In this paper, we study the bound- edness of the Marcinkiewicz operator associated with Schrodinger operator μ L on generalized Morrey spaces Mp,ϕ .W efind the sufficient conditions on the pair (ϕ1,ϕ2) which ensures the boundedness of the operators μ L from one generalized Morrey space Mp,ϕ1 to another Mp,ϕ2 , 1 < p < ∞ and from the space M1,ϕ1 to the weak space WM1,ϕ2 .