On the index of pseudo-differential operators on compact Lie groups

In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean-Singer index formula together with the operator-valued functional calculus developed here

A note on the Fourier transform in Hölder spaces

En este artículo, se estudia la acotación de la transformada periódica de Fourier desde espacios de Lebesgue a Espacios Hölder. Par- ticularmente, se generaliza un resultado clásico de Bernstein.

Multilinear analysis for discrete and periodic pseudo-differential operators in Lp-spaces

In this note we announce our investigation on the Lp properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators byshowing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentialsas well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.En esta nota anunciamos los resultados de nuestra investigación sobre las propiedades Lp de operadores pseudodiferenciales multilineales periódicos y/o discretos. Primero, revisaremos el análisis multilineal de tales operadores mostrando versiones análogas de los teoremas clásicos disponibles en el análisis multilineal euclidiano (debidos a Coifman y Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.), pero, en el contexto de operadores periódicos y/o discretos. Se caracterizará la s-nuclearidad, 0 < s ≤ 1, para operadores multilineales pseudodiferenciales periódicos y/o discretos. Para cumplir este objetivo se clasificarán aquellos operadores lineales s-nucleares, 0 < s ≤ 1, multilineales con núcleo, sobre espacios de Lebesgue arbitrarios definidos en espacios de medida σ-finitos. Finalmente, como aplicación de los resultados presentados se obtiene la versión periódica de la desigualdad de Kato-Ponce, y se examina la s-nuclearidad de potenciales de Bessel lineales y multilineales, como también la s-nuclearidad de operadores integrales de Fourier periódicos admitiendo símbolos con tipos adecuados de singularidad

Weak type (1,1) bounds for a class of periodic pseudo-differential operators

In this work we establish the weak (1,1) continuity for pseudo-differential operators with symbols in toroidal (1, delta) classes

Nuclear pseudo-differential operators in Besov spaces on compact Lie groups

In this work we establish the metric approximation property for Besov spaces defined on arbitrary compact Lie groups. As a consequence of this fact, we investigate trace formulae for nuclear Fourier multipliers on Besov spaces. Finally, we study the r-nuclearity, the Grothendieck-Lidskii formula and the (nuclear) trace of pseudo-differential operators in generalized Hormander classes acting on periodic Besov spaces. We will restrict our attention to pseudo-differential operators with symbols of limited regularity

Besov continuity for pseudo-differential operators on compact homogeneous manifolds

In this paper we study the Besov continuity of pseudo-differential operators on compact homogeneous manifolds M = G/K. We use the global quantization of these operators in terms of the representation theory of compact homogeneous manifolds

Hölder-Besov boundedness for periodic pseudo-differential operators

In this work we give Holder-Besov estimates for periodic Fourier multipliers. We present a class of bounded pseudo-differential operators on periodic Besov spaces with symbols of limited regularity

Invertibility for a class of Fourier multipliers

In this paper, we establish invertibility for a class of multipliers in the setting of Hormander quantization of pseudo-differential operators on R-n More precisely, the existence of inverses and fundamental solutions of these operators are investigated

Pseudo-differential operators on ℤⁿ with applications to discrete fractional integral operators

In this manuscript we provide necessary and sufficient conditions for the weak( 1, p) boundedness, 1 < p < 8, of discrete Fourier multipliers ( Fourier multipliers on Zn). Our main goal was to apply the results obtained to discrete fractional integral operators. Discrete versions of the Calderon- Vaillancourt theorem and the Gohberg lemma also are proved