Journal of Pseudo-Differential Operators and Applications最新文献

We introduce the directional short-time fractional Fourier transform (DSTFRFT) and prove an extended Parseval’s identity and a reconstruction formula for it. We also investigate the continuity of both the directional short-time fractional Fourier transform and its synthesis operator on the appropriate space of test functions. Using the obtained continuity results, we develop a distributional framework for the DSTFRFT on the space of tempered distributions (mathcal {S}'(mathbb {R}^n)). We end the article with a desingularization formula.

In this paper, we give necessary and sufficient conditions for the boundedness of Hausdorff operator on power-weighted Morrey–Herz spaces that are associated with the Opdam–Cherednik transform. Moreover, we provide some Lipschitz estimates for its commutator on such spaces.

Let X be a ball quasi-Banach function space on ({mathbb {R}}^{n}) and (h_{X}({mathbb {R}}^{n})) the local Hardy space associated with X. In this paper, under some reasonable assumptions on both X and another ball quasi-Banach function space Y, we aim to derive the boundedness of pseudo-differential operators with symbols in (S^{-alpha }_{1,delta }) from (h_{X}({mathbb {R}}^{n})) to (h_{Y}({mathbb {R}}^{n})) via applying the extrapolation theorem. In order to prove this result, we also establish the infinite and finite atomic decompositions for the weighted local Hardy space (h^{p}_{omega }({mathbb {R}}^{n})) and obtain the mapping property of the above pseudo-differential operators from (h^{p}_{omega ^{p}}({mathbb {R}}^{n})) to (h^{q}_{omega ^{q}}({mathbb {R}}^{n})). Moreover, the above results have a wide range of generality. For example, they can be applied to the variable Lebesgue space, the Lorentz space, the mixed-norm Lebesgue space, the local generalized Herz space and the mixed Herz space.

The boundedness of bilinear Fourier integral operators with certain non-degenerate phase functions is proved, which is a bilinear version of Seeger, Sogge, and Stein’s theorem concerning the (L^p) boundedness of Fourier integral operators. Our result gives an improvement of the result of Rodríguez-López, Rule, and Staubach proved in 2014.

We study a class of Hardy-Sobolev equations containing both sign-changing and negative power terms on closed Riemannian manifolds. With the help of a modified Nehari manifold method and some variational techniques, the existence and multiplicity of positive weak solutions are established, along with blow-up behavior analysis.

The present paper aims to establish the existence of at least two weak solutions of a nonlocal singular problem governed by a generalized integro-differential operator with singular kernel in a bounded domain (Omega ) of (mathbb {R}^N) with Lipschitz boundary. The main variational tool is based on the Nehari manifold approach and the fibering maps analysis. Moreover, we state and prove two embedding results of the generalized fractional Sobolev spaces into generalized weighted Lebesgue spaces, which serve as pivotal components in our principal proof.

In this paper, we consider an inverse problem related to the fractional pseudo-parabolic equation with a nonlinear source term. Our investigation reveals the ill-posedness of the problem according to Hadamard’s definition. We present two improved variations of the optimal filtering method introduced by Seidman (SIAM J Numer Anal 33:162–170, 1996) to establish some optimal estimates under some an a priori assumptions on the regularity of the exact solution. Finally, the effectiveness of our algorithm is demonstrated through numerical examples.

In this note, we deal with the fractional logarithmic Schrödinger operator ((I+(-Delta )^s)^{log }) and the corresponding energy spaces for variational study. The fractional (relativistic) logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, (log (1+|xi |^{2s})), (s>0). We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of ({mathbb {R}}^N.) We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.

Our main focus in this paper is to explore the mapping properties for the n-dimensional fractional Hausdorff operator (H_{Phi ,beta }) from (L^{p}(mathbb {R}^{n},|x|^{alpha })) to (L^{q}(mathbb {R}^{n},|x|^{gamma })), where (p,q<1~(p,qne 0)), (alpha ,gamma in mathbb {R}), (0le beta <n) and (Phi ) is a nonnegative measurable function on (mathbb {R}^n). For (p,q<1~(p,qne 0)) satisfying some additional assumptions, we give sufficient conditions for the validity of the reverse fractional Hausdorff inequality (left| H_{Phi ,beta }fright| _{L^{q}(mathbb {R}^{n},|x|^{gamma })}ge CVert fVert _{L^{p}(mathbb {R}^{n},|x|^{alpha })}) for some positive constant C and all nonnegative functions (fin L^{p}(mathbb {R}^{n},|x|^{alpha })). For the particular case (0<p=q<1), we obtain the sharp reverse fractional Hausdorff inequality. As applications, we establish the sharp reverse inequalities for the n-dimensional fractional Hardy operator and its adjoint operator, and also the n-dimensional fractional Hardy–Littlewood–Pólya operator on power-weighted Lebesgue spaces.

In an effort to express Clifford-valued signals efficiently in time–frequency domain, we introduce the notion of the novel integral transform known as Clifford-valued linear canonical wave-packet transform (CLCWPT). In the beginning, we derived the fundamental properties of the proposed transform which include linearity, anti-linearity, scaling parity, dilation and Parseval’s formula. Moreover, some important signal analysis results have been established, viz. energy conservation, inversion formula, characterization of range and bounds of clifford valued linear canonical wave-packet transform. We culminate our manuscript by studying corresponding Heisenberg’s uncertainty principle and logarithmic uncertainty principle associated with CLCWPT.