Two-weighted estimates for some sublinear operators on generalized weighted Morrey Spaces and applications

Fuente: arXiv

Guardado en:

Detalles Bibliográficos
Autores principales:	Ramadana, Yusuf, Gunawan, Hendra
Formato:	Preprint
Publicado:	2024
Materias:	Functional Analysis 42B25, 42B20
Acceso en línea:	Acceder al recurso
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

_version_	1866916280160747520
author	Ramadana, Yusuf Gunawan, Hendra
author_facet	Ramadana, Yusuf Gunawan, Hendra
contents	In this paper we investigate the boundedness of sublinear operators generated by fractional integrals as well as sublinear operators generated by Calderòn-Zygmund operators on generalized weighted Morrey spaces and generalized weighted mixed-Morrey spaces. In particular we are interested in the strong-type estimate for $1<p<\infty$ and the weak estimate for $p=1$. Under some assumptions, we prove that the operators and their commutator with a BMO function are bounded on those function spaces with different weights. The results then imply the boundedness of fractional integrals with Gaussian kernel bounds as well as with rough kernels, fractional maximal integrals with rough kernels, and sublinear operators with rough kernels generated by Calderòn-Zygmund operators. Using the results, we obtain some regularity properties of the solution of some partial differential equations.
format	Preprint
id	arxiv_https___arxiv_org_abs_2406_05435
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	Two-weighted estimates for some sublinear operators on generalized weighted Morrey Spaces and applications Ramadana, Yusuf Gunawan, Hendra Functional Analysis 42B25, 42B20 In this paper we investigate the boundedness of sublinear operators generated by fractional integrals as well as sublinear operators generated by Calderòn-Zygmund operators on generalized weighted Morrey spaces and generalized weighted mixed-Morrey spaces. In particular we are interested in the strong-type estimate for $1<p<\infty$ and the weak estimate for $p=1$. Under some assumptions, we prove that the operators and their commutator with a BMO function are bounded on those function spaces with different weights. The results then imply the boundedness of fractional integrals with Gaussian kernel bounds as well as with rough kernels, fractional maximal integrals with rough kernels, and sublinear operators with rough kernels generated by Calderòn-Zygmund operators. Using the results, we obtain some regularity properties of the solution of some partial differential equations.
title	Two-weighted estimates for some sublinear operators on generalized weighted Morrey Spaces and applications
topic	Functional Analysis 42B25, 42B20
url	https://arxiv.org/abs/2406.05435

if (filemtime($filename) + $maxlifetime < time()) { unlink($filename); $count++; } } return $count; } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ protected function saveSession($sessId, $data): bool { $sessFile = $this->path . '/sess_' . $sessId; if ($handle = fopen($sessFile, 'w')) { $return = false; // Lock the file for exclusive access to avoid issues with multiple // processes writing session simultaneously: if (flock($handle, LOCK_EX)) { $return = fwrite($handle, $data); // Make sure that there's no trailing data by truncating the file to // the correct length: ftruncate($handle, strlen($data)); fflush($handle); flock($handle, LOCK_UN); } fclose($handle); if ($return !== false) { return true; } } // If we got this far, something went wrong with the file output... // It is tempting to throw an exception here, but this code is called // outside of the context of exception handling, so all we can do is // echo a message.

// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
language	"es"
ui	"standard"
VUFIND_SESSION	"4f12572a79150d70bacc6c853546430f"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1791203103.2519 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "4f12572a79150d70bacc6c853546430f" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1762}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"language=es; ui=standard; VUFIND_SESSION=4f12572a79150d70bacc6c853546430f"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.217.58"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"51743"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2406_05435/Details """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2406_05435/Details"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1791203103.2474
REQUEST_TIME	1791203103

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
language	"es"
ui	"standard"
VUFIND_SESSION	"4f12572a79150d70bacc6c853546430f"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1791203103.2519 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "4f12572a79150d70bacc6c853546430f" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1762}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"language=es; ui=standard; VUFIND_SESSION=4f12572a79150d70bacc6c853546430f"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.217.58"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"51743"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2406_05435/Details """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2406_05435/Details"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1791203103.2474
REQUEST_TIME	1791203103

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Ejemplares similares

Environment & details:

Environment & details: