A framework of linear canonical Hankel transform pairs in distribution spaces and their applications

The motivation of this article stems from the fact that weak solutions of some partial differential equations exist in a distributional sense, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. For $$1\leqq p<\infty $$ and $$s\in {{\mathbb {R}}}$$ , we have introduced a new definition for each of the following Sobolev-type spaces: $$\begin{aligned} {\mathscr {W}}^{s,p,{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }(I) \qquad \text {and} \qquad {\mathscr {W}}^{s,p,{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }(I) \end{aligned}$$ as subspaces of $$\begin{aligned} H^{'{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }(I) \qquad \text {and} \qquad H^{'{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }(I), \end{aligned}$$ respectively, by using a linear canonical Hankel transform pair, where $$ \mu ,\nu ,\alpha $$ and $$\beta $$ are real parameters and $${\mathscr {M}}$$ is a $$2\times 2$$ real (or complex) matrix with determinant equal to 1. Any $$f\in {H^{'{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }}(I)$$ and $$g\in {H^{'{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }}(I)$$ with compact support are shown to be an element of the spaces: $$\begin{aligned} {\mathscr {W}}^{s,p,{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }(I) \qquad \text {and} \qquad {\mathscr {W}}^{s,p,{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }(I), \end{aligned}$$ respectively, for the large negative value of s. Examples in these spaces are constructed and the corresponding solutions are obtained. We have shown that these spaces turn out to be Hilbert spaces with respect to a certain norm with the dual spaces: $$\begin{aligned} {\mathscr {W}}^{-s,p,{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }(I) \qquad \text {and} \qquad {\mathscr {W}}^{-s,p,{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }(I), \end{aligned}$$ respectively. Further, if $$f\in {{\mathscr {W}}^{s,p,{\mathscr {M}}}_{1,\mu ,\nu ,\alpha ,\beta }(I)}$$ , then $$x^{-\nu \mu +\alpha -2\nu +1}f(x)$$ is shown to be bounded. Similarly, if $$g\in {{\mathscr {W}}^{s,p,{\mathscr {M}}}_{2,\mu ,\nu ,\alpha ,\beta }(I)}$$ , then $$x^{-\nu \mu -\alpha }g(x)$$ is also shown to be bounded. Furthermore, some applications of linear canonical Hankel transform pairs are provided in order to solve some generalized non-homogeneous partial differential equations. Finally, in the concluding section, some motivations and directions are indicated for further researches related to the areas which are considered and discussed in this article.