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This paper presents the properties of Gyrator potential operator $ \mathcal {G}_{\mu }^{l} $ G μ l and related $ L^p $ L p -Sobolev spaces associated with Gyrator transform (GT). The Schwartz-type space $ S_{\Delta }(\mathbb {R}^2) $ S Δ (R 2) is introduced. Various properties of the kernel of GT is studied including its continuity. Pseudo-differential operators $ \boldsymbol {\mathcal {A}_{\alpha,q}} $ A α , q associated with GT is defined and derived it's continuity on $ S_{\Delta }(\mathbb {R}^2) $ S Δ (R 2). The Gyrator potential operator $ \mathcal {G}_{\mu }^{l} $ G μ l is defined as a pseudo-differential operator associated with a precise symbol and obtained certain fruitful properties. The operator $ \mathcal {G}_{\mu }^{l} $ G μ l is extended to a space of distributions. Another integral representation of $ \boldsymbol {\mathcal {A}_{\alpha,q}} $ A α , q is obtained and its $ L^p(\mathbb {R}^2) $ L p (R 2) -boundedness result is derived. The spaces $ \mathcal {H}_{\alpha }^{m,p}(\mathbb {R}^2) $ H α m , p (R 2) and $ H_{\Delta }^{m,p}(\mathbb {R}^2) $ H Δ m , p (R 2) are defined. It is shown that the operator $ \mathcal {G}_{\mu }^{l} $ G μ l is an isometry of $ H_{\Delta }^{m,p}(\mathbb {R}^2) $ H Δ m , p (R 2). An $ L^p $ L p -boundedness result for the operator $ \mathcal {G}_{\mu }^{l} $ G μ l is proved. We conclude the article by applying some of the results to show that the analytical solution of certain non-homogeneous partial differential equations belong to these spaces. [ABSTRACT FROM AUTHOR]

In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator on some ultradifferentiable function spaces. [ABSTRACT FROM AUTHOR]

This article is concerned with the study of the continuity of wavelet transform involving fractional Hankel transform on certain function spaces. Then-dimensional boundedness property of the fractional wavelet transform is also discussed on Sobolev type space. Particular cases are also considered. [ABSTRACT FROM PUBLISHER]