arXiv subjects
Publications and source records attributed to Bojan Prangoski.
We prove the complex powers of a class of infinite order hypoelliptic pseudodifferential operators can always be represented as hypoelliptic pseudodifferential operators modulo ultrasmoothing operators. We apply this result to the study of semigroups generated by square roots of non-negative hypoelliptic infinite order operators. For this purpose, we derive precise estimates of the corresponding heat kernel.
We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.
We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors $\mathcal{O}'^{\ast}_{C}(\mathbb{R}^{d})$ for tempered ultradistributions are analyzed via the duality with respect to the test function spaces $\mathcal{O}^{\ast}_{C}(\mathbb{R}^{d})$, introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions $T,S\in \DD'^{\{M_p\}}\left(\RR^d\right)$ exists if and only if $\left(φ*\check{S}\right)T\in\DD'^{\{M_p\}}_{L^1}\left(\RR^d\right)$ for every $φ\in\DD^{\{M_p\}}\left(\RR^d\right)$.
We study the expansions of the elements in $\mathcal S(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ with respect to the Laguerre orthonormal basis, extending the result of M. Guillemont-Teissier in the case $d=1$. As a consequence, we obtain the Schwartz kernel theorem for $\mathcal{S}(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ and the extension theorem of Whitney type for $\mathcal{S}(\mathbb{R}_+^d)$.
The first part of the paper is devoted to the $G$-type spaces i.e. the spaces $G^α_α(\mathbb R^d_+)$, $α\geq 1$ and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justification of obtained results. The Laguerre type expansions of the elements in $G^α_α(\mathbb R^d_+)$, $α\geq 1$ and their duals characterise these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological description and by the nuclearity of $G_α^α(\mathbb{R}^d_+)$, $α\geq 1$ the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the $G$-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.
A class of translation-invariant Banach spaces of quasianalytic ultradistributions is introduced and studied. They are Banach modules over a Beurling algebra. Based on this class of Banach spaces, we define corresponding test function spaces $\mathcal{D}^*_E$ and their strong duals $\mathcal{D}'^*_{E'_{\ast}}$ of quasianalytic type, and study convolution and multiplicative products on $\mathcal{D}'^*_{E'_{\ast}}$. These new spaces generalize previous works about translation-invariant spaces of tempered (non-quasianalytic ultra-) distributions; in particular, our new considerations apply to the settings of Fourier hyperfunctions and ultrahyperfunctions. New weighted $\mathcal{D}'^{\ast}_{L^{p}_η}$ spaces of quasianalytic ultradistributions are analyzed.
We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of $u$ with sub-exponential growth with respect to $u,$ we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution $u$.
We construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces.
Results of Da Prato and Sinestrari, on differential operators with non-dense domain but satisfying the Hille--Yosida condition, are applied in the setting of Beurling weighted spaces of ultradistributions $\DD'^{(s)}_{L^p}((0,T)\times U)$, where $U$ is open and bounded set in $\mathbb R^d$.
Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic calculus is developed.
The Laplace transform in Komatsu ultradistributions is considered. Also, conditions are given under which an analytic function is a Laplace transformation of an ultradistribution.
The equivalence of several definitions of convolution of two Roumieu ultradistributions is proved. For that purpose, the $\varepsilon$ tensor product of $\dot{\tilde{\mathcal{B}}}^{\{M_p\}}$ and a locally convex space $E$ is considered.
The connection between the Anti-Wick and Weyl quantization is given for certain class of global symbols, which corresponding pseudodifferential operators act continuously on the space of tempered ultradistributions of Beurling, respectively, of Roumieu type. The largest subspace of ultradistributions is found for which the convolution with the gaussian kernel exist. This gives a way to extend the definition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.