Journal of Pseudo-Differential Operators and Applications最新文献

In this paper, we prove a potentially useful (L^p(dnu ))-boundedness result for the pseudo-differential operators associated with the spherical mean operator. Also, boundedness result for symmetrically global pseudo-differential operator on (L^p(dnu ))-type Sobolev space (mathcal {H}^{u,v,p}) of order (u, v) are discussed. An application in solving a generalized heat equation is given.

This paper is devoted in investigations concerning the study of the coupled potential operator (J_{s}^{alpha , beta }) and corresponding (L^p)-Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space (mathcal {S}_{alpha ,beta }) 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}^{alpha , beta }) is defined as a pseudo-differential operator related with a precise symbol. The operator (J_{s}^{alpha , beta }) is extended to a space of distributions. An (L^p)-Sobolev boundedness result for the operator (J_{s}^{alpha , beta }) is shown. The spaces (H^{m,alpha ,beta }_{p}) and (mathcal {H}^{m,alpha ,beta }_{p}) introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces.

In this paper, we introduce a new space that generalizes the (phi )-Hilfer space with the (xi (cdot ))-Laplacian operator, denoted ((phi ,{xi }(cdot )))-HFDS. We refer to this new space as the (phi )-fractional space with anisotropic (overrightarrow{xi }(cdot ))-Laplacian operator, abbreviated as ((phi ,overrightarrow{xi }(cdot )))-HFDAS. We prove that ((phi ,overrightarrow{xi }(cdot )))-HFDAS is a separable, and reflexive Banach space. Furthermore, we extend some well-known properties and embedding results of the ((phi ,xi (cdot )))-HFDS space to ((phi ,overrightarrow{xi }(cdot )))-HFDAS. Moreover, we illustrate an application of ((phi ,overrightarrow{xi }(cdot )))-HFDAS by solving a differential equation via variational methods.

This paper introduces new classes of p-adic operators representing m-dissipative pseudo-differential operators with negative definite symbols under certain conditions. We will study new types of semilinear problems and martingale problems associated with these operators, and we will prove that these pseudo-differential operators are the infinitesimal generators of strongly continuous contraction semigroups on (L^2({mathbb {Q}}_p^n)). Also, this article introduces new families of measures, resolvent of measures, positive definite measures, Feller semigroups, and Markov processes.

This study investigates the relationship between the moduli of continuity of a function and its Hartley transform. We explore this connection by deriving significant results such as the Riemann–Lebesgue lemma, Parseval’s theorem, and the Hausdorff–Young inequality for the Hartley transform in both the Euclidean space and torus. Using a translation operator, we obtain an analog of Titchmarsh’s theorem for the Hartley transform. In addition, we extend our analysis to the Hartley series on the torus.

Recently, wavelet analysis over the p-adic fields are widely used in physics, biology and geophysics. In this paper, p-adic wavelets are used to study various p-adic pseudo-differential equations. p-Adic analogue of wave equation and general linear second order pseudo-differential equation are solved using both Haar and non-Haar wavelets. Finally, the existence of solutions of nonlinear p-adic pseudo differential equation and p-adic analogue of Navier Stokes equation are proved using the Schauder fixed point theorem together with wavelet functions.

We report that the quantity measured in the (L^2) norm of the solution itself of the generalized Rosenau equation, which was completely unknown in this equation, grows in the proper order at time infinity. It is also immediately apparent that this growth aspect does not occur in three or more spatial dimensions, so we will apply the results obtained in this study to provide another proof that Hardy-type inequalities do not hold in the case of one or two spatial dimensions.

In this paper, we present various inequalities regarding the linear combinations of orthogonal projections. These results aim to generalize and refine well-known inequalities, such as those due to Buzano and Ostrowski. Additionally, we investigate a specific case of these linear combinations and introduce new refinements of the Cauchy–Schwarz inequality. Furthermore, we establish some findings related to the covariance and variance of bounded linear operators. Moreover, as applications of some of our results, we establish several inequalities involving the product of three operators, one of which is a linear combination of an orthogonal projection and the identity operator. Finally, we introduce a new positive operator construction in terms of an orthogonal projection and the identity operator, and we derive some norms and numerical radius inequalities involving it.

In the first part we analyze space ({mathcal {G}}^*({mathbb {R}}^{n}_+)) and its dual through Laguerre expansions when these spaces correspond to a general sequence ({M_p}_{pin {mathbb {N}}_0}), where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form (Lu=f,; L=sum _{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),) where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on ({mathbb {R}}^n_+) and ({mathbb {R}}^m); (A_j, j=1,...,k), (E_y) and (P(x,D_x)) are operators whose eigenfunctions form orthonormal basis of corresponding (L^2-)space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.

In this article we study various forms of the Hardy inequality for affine connections on a complete noncompact Riemannian manifold, including the two-weight Hardy inequality, the improved Hardy inequality, the Rellich inequality, the Hardy–Poincaré inequality and the Heisenberg–Pauli–Weyl inequality. Our results improve and include many previously known results as special cases.