Weighted boundedness of multilinear integral operators for the endpoint cases

<abstract><p>We prove the weighted boundedness for the multilinear operators associated to some integral operators for the endpoint cases. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.</p></abstract>

Bochner–Riesz operators in grand lebesgue spaces

We provide the conditions for the boundedness of the Bochner–Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate for the constant appearing in the Lebesgue–Riesz norm estimation of the Bochner–Riesz operator and we investigate the convergence of the Bochner–Riesz approximation in Lebesgue–Riesz spaces.

Operadores de riesz en el Alglat(T)∩{T}

En este trabajo X es un espacio de Banach y B(X) denota los operadores acotados. Si T∈B(X), por lat(T) entenderemos los subespacios invariantes por T. Se dice que T es lleno, si (T(M)) ̅=M, para todo M∈lat(T) (la barra indica la clausura en la topología inducida por la norma). Se prueba principalmente el siguiente resultado: Sean X un espacio de Banach y T ∈B(X) acotado por abajo. Sea K ∈Alglat(T)∩{T}' un operador de Riesz. Si K es lleno, entonces T es lleno. Aquí Alglat(T)={S∈B(X):M∈lat(T)⟾M∈lat(S)} y {T}^'={S∈B(X):S∘T=T∘S}. Palabras clave: Operador lleno, operador de Riesz, operador acotado por abajo. Abstract In this work X is a Banach space and B(X) denotes the bounded operators. If T ∈B(X), for lat(T) we will understand the invariant subspaces for T. An operator T is full, if (T(M)) ̅=M, for all M∈ latT (the bar indicates the closure in the topology induced by the norm). The following result is true: Let X be a Banach space, T ∈B(X) a bounded below operator and K ∈Alglat(T)∩{T}' a Riesz operator: If K is a full operator, then T is a full operator. Here Alglat(T)={S∈B(X):M∈lat(T)⟾M∈lat(S)} and {T}^'={S∈B(X):S∘T=T∘S}. Keywords: full operator, Riesz operator, bounded below operator.

RESULTADOS SOBRE OPERADORES LLENOS EN ESPACIOS DE HILBERT

Se prueba, entre otros, el siguiente resultado: Sea T:H→H un operador autoadjunto inyectivo, y K:H→H un operador de Riesz, tal que K∈Alglat(T)∩{T}'. Si K:H→H es lleno, entonces T:H→H es lleno. Palabras clave: Operador de Riesz, operador autoadjunto, operador lleno. Abstract It is proved here, among other results, the following: Let T:H→H be a self-adjoint injective operator, and K:H→H a Riesz operator, such that K∈Alglat(T)∩{T}'. If K:H→H is a full operator, then T:H→H is a full operator. Keywords: Riesz operator, self-adjoint operator, full operator. Resumo O siguiente resultado, entre outros, está provado: Seja T:H→H um operador autoadjunto limitado abaixo, e K:H→H um operador de Riesz, tal qual K∈AlglatT⋂{T}^'. Se K:H→H é um operador completo, então T:H→H é um operador completo. Palavras-chave: operador Riesz, operador autoadjunto completo.

Mittag–Leffler Memory Kernel in Lévy Flights

In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.