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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] more...

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings. [ABSTRACT FROM AUTHOR] more...

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo‐differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo‐differential operators are derived. It is shown that pseudo‐differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation. [ABSTRACT FROM AUTHOR] more...

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation. [ABSTRACT FROM AUTHOR] more...

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] more...