The present paper is in continuation to authors earlier paper {9} where two variable analogues of certain fractional integral operators of M. Saigo were investigated. This paper deals with the effect of operating two variable analogues of Mellin and Laplace transforms on these two variable analogues of fractional integral operators of the earlier paper.

The Paper deals with a two variable analogues of certain fractional integral operators introduced by M. Saigo. Besides giving two variable analogues of earlier known fractional integral operators of one variable as special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.

En los ultimos 10 a~nos importantes avances han sido obtenidos en el problema de dos pesos del tipo d ebil (1; 1) de operadores de Calder on-Zygmund. Todos esos resultados est an asociados a lo que se conoce como conjetura de Muckenhoupt-Wheeden. Por otro lado, tambi en recientemente muchos avances han sido obtenidos en variantes cuantitativas de desigualdades pesadas de ciertos operadores cl asicos del an alisis arm onico. Una de las herramientas claves que ha aparecido para estos avances es lo que se conoce como dominaci on por operadores sparse de los operadores de Calder on-Zygmund y variantes. En esta tesis avanzamos en estas dos l neas. Por un lado encontramos resultados negativos para una variante m as d ebil de la conjetura de Muckenhoupt-Wheeden. Y por otro lado obtenemos varias estimaciones cuantitativas en desigualdades de tipo d ebil mixto para varios operadores, como las integrales singulares y los conmutadores de dichos operadores. La principal novedad de estas estimaciones es que por primera vez se usa la herramienta de dominaci on sparse en este contexto. / In the last ten years very important advances related to the endpoint two-weight weak type inequality of Calderón-Zygmund operators and Commutators are been obtained. On the other hand relevant advances in quantitative variant of several weighted estimates are been showed. A key tool for such advances have been the sparse domination of these classical operators. In this work we advance in these two lines. In rst place we show negative results in a variant of the Muckenhoupt-Wheeden Conjecture related to the two-weight weak type (1; 1) estimate of the Hilbert transform. On the other hand, we obtain quantitative estimates in the called weighted weak mixed-type estimates for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this part of the work lies in the fact that we rely upon sparse domination results for this kind of inequalities