scholarly journals Sobolev type inequalities in ultrasymmetric spaces with applications to Orlicz-Sobolev embeddings

2005 ◽  
Vol 3 (2) ◽  
pp. 183-208 ◽  
Author(s):  
Evgeniy Pustylnik
Keyword(s):  
Function Space ◽  
Sobolev Spaces ◽  
Banach Function Space ◽  
Sobolev Type ◽  
Small Space ◽  
Partial Derivatives ◽  
Sobolev Embeddings ◽  
Derivatives Of

LetDkfmean the vector composed by all partial derivatives of orderkof a functionf(x),x∈Ω⊂ℝn. Given a Banach function spaceA, we look for a possibly small spaceBsuch that‖f‖B≤c‖|Dkf|‖Afor allf∈C0k(Ω). The estimates obtained are applied to ultrasymmetric spacesA=Lφ,E,B=Lψ,E, giving some optimal (or rather sharp) relations between parameter-functionsφ(t)andψ(t)and new results for embeddings of Orlicz-Sobolev spaces.

scholarly journals Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Zhiyong Wang ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Sobolev Spaces ◽  
Equivalence Relation ◽  
Fractional Derivatives ◽  
Sobolev Type ◽  
Square Function ◽  
Heisenberg Groups ◽  
Square Functions ◽  
Regularity Properties ◽  
Reverse Hölder ◽  
Derivatives Of

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

On the Regularity of the Multisublinear Maximal Functions

2015 ◽  
Vol 58 (4) ◽  
pp. 808-817 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Sobolev Spaces ◽  
Maximal Function ◽  
Maximal Operator ◽  
Maximal Functions ◽  
Partial Derivatives ◽  
First Order ◽  
Derivatives Of

AbstractThis paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Partial Derivatives of the Lambert Problem

2014 ◽  
Author(s):  
Nitin Arora ◽  
Ryan P. Russell ◽  
Nathan J. Strange
Keyword(s):  
Partial Derivatives ◽  
Lambert Problem ◽  
Derivatives Of

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

scholarly journals ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

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