Abstract
We define the weighted Dirichlet integral and show that this integral can be represented by a multidimensional generalized shift. The corresponding norm does not allow us to define the function spaces of arbitrary fractional order of smoothness, and so we introduce the new norm that is related to a generalized Bessel potential. Potential theory originates from the theory of electrostatic and gravitational potentials and the study of the Laplace, wave, Helmholtz, and Poisson equations. The celebrated Riesz potentials are the realizations of the real negative powers of the Laplace and wave operators. In the meantime, much attention in potential theory is paid to the Bessel potential generating the spaces of fractional smoothness. We progress in generalization by considering the Laplace–Bessel operator constructed from the singular Bessel differential operator. The theory of singular differential equations with the Bessel operator as well as the theory of the corresponding weighted function spaces are closely connected and belong to the areas of mathematics whose theoretical and applied significance can hardly be overestimated.
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1. Introduction
Well known is the fundamental role of the classical Bessel potentials in the general theory of function spaces of fractional smoothness as well as applications to the theory of partial equations (see [1, 2]). The classical Bessel potentials are constructed on using the Fourier transform.
The goal of this article is to develop the theory of the space of generalized Bessel potentials \( \mathbf{B}^{\alpha}_{\gamma} \) constructed with the Hankel transform. This space was first introduced by Lyakhov in [3] who based on the Stein–Lizorkin approach. In [3], the \( B \)-hypersingular integrals and \( B \)-Riesz potentials, introduced earlier by Lyakhov in [4, 5], were applied the constructing the norm on \( \mathbf{B}^{\alpha}_{\gamma} \). In the present article, we use another approach to \( \mathbf{B}^{\alpha}_{\gamma} \) which is based on the works [6,7,8] by Aronszajn and Smith. The approach consists in introducing some norm on \( \mathbf{B}^{\alpha}_{\gamma} \) with the help of weighted Dirichlet integrals.
The spaces of generalized Bessel potentials of arbitrary order \( \alpha \) are necessary for defining the classes of solutions to the boundary value problem
where \( A \) is an elliptic operator containing the Bessel differential operators \( B_{\gamma_{i}}=\frac{\partial^{2}}{\partial x_{i}^{2}}+\frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} \); in particular, \( A \) can be the Laplace–Bessel operator \( \Delta_{\gamma}=\sum_{i=1}^{n}B_{\gamma_{i}} \).
2. Preliminaries
Let \( ^{n} \) be the \( n \)-dimensional Euclidean space, while
Let \( \gamma=(\gamma_{1},\dots,\gamma_{n}) \) be the multi-index consisting of fixed positive reals \( \gamma_{i} \), with \( i=1,\dots,n \), and \( |\gamma|=\gamma_{1}+\dots+\gamma_{n} \).
Let \( \Omega \) be a finite or infinite open set in \( ^{n} \) symmetric with respect to each of the hyperplanes \( x_{i}=0 \), with \( i=1,\dots,n \), \( \Omega_{+}=\Omega\cap{}^{n}_{+} \), and \( \overline{\Omega}_{+}=\Omega\cap\overline{}^{n}_{+} \). We will work with the function class \( C^{m}(\Omega_{+}) \) consisting of \( m \) times differentiable functions on \( \Omega_{+} \). Denote by \( C^{m}(\overline{\Omega}_{+}) \) the subset of the functions in \( C^{m}(\Omega_{+}) \) whose all derivatives with respect to \( x_{i} \) extend continuously to \( x_{i}=0 \) for all \( i=1,\dots,n \). Let \( C^{m}_{ev}(\overline{\Omega}_{+}) \) consist of the functions in \( C^{m}(\overline{\Omega}_{+}) \) such that
for all nonnegative integers \( k\leq\frac{m-1}{2} \) (see [9, p. 21]). In what follows, we use the abbreviation \( C^{m}_{ev} \) for \( C^{m}_{ev}(\overline{}^{n}_{+}) \) and put
where the intersection is taken over all finite \( m \) and \( C^{\infty}_{ev}(\overline{}_{+})=C^{\infty}_{ev} \).
Let \( {\overset{\circ}{C}}\,^{\infty}_{ev}(\,\overline{\Omega}_{+}) \) be the space of compactly supported functions \( f\in C^{\infty}_{ev}(\,\overline{\Omega}_{+}) \). Put
Let \( L_{p}^{\gamma}({}^{n}_{+})=L_{p}^{\gamma} \), with \( 1\leq p<\infty \), consists of the measurable functions on \( {}^{n}_{+} \) even in each of the variables \( x_{i} \), with \( i=1,\dots,n \), such that
Here and in the sequel \( x^{\gamma}=\prod_{i=1}^{n}x_{i}^{\gamma_{i}} \). For every \( p\geq 1 \), the norm of \( f\in L_{p}^{\gamma} \) is defined as
It is know that \( L_{p}^{\gamma} \) is a Banach space (see [9]).
The multidimensional Hankel transform of \( f\in L^{\gamma}_{1}(^{n}_{+}) \) is defined as
where
Denote by \( j_{\nu} \) the normalized Bessel function of the first kind \( j_{\nu}(x)=\frac{2^{\nu}\Gamma(\nu+1)}{x^{\nu}}J_{\nu}(x) \), where \( J_{\nu} \) is a Bessel function of the first kind (see [10]).
Let \( f\in L_{1}^{\gamma}(_{+}) \) be a function of bounded variation in a neighborhood of a continuity point \( x \) of \( f \). Then for \( \gamma>0 \) the inversion formula for the transformation looks as
The Hankel transform reduces the Bessel operator to the multiplication of the square of the corresponding argument with the minus sign (see [9]):
where \( (B_{\gamma_{i}})_{x_{i}}=\frac{\partial^{2}}{\partial x_{i}^{2}}+\frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} \) is the Bessel operator, \( i=1,\dots,n \).
Parseval’s identity for the one-dimensional Hankel transform (see [9, p. 20]) and the fact that \( f\in L_{2}^{\gamma}(^{n}_{+}) \) yield \( \mathbf{F}_{\gamma}f\in L_{2}^{\gamma}(^{n}_{+}) \) and
The multidimensional generalized shift is defined as
where each one-dimensional generalized shift \( {\,{}^{\gamma_{i}}T_{x_{i}}^{y_{i}}} \), \( i=1,\dots,n \), acts by the formula
In what follows, put
The generalized convolution generated by the multidimensional generalized shift \( {}^{\gamma}\mathbf{T}^{y}_{x} \) has the form
The multidimensional Poisson operator \( \mathbf{P}^{\gamma}_{x} \) acts an integrable function \( f \) by the formula
3. The Weighted Dirichlet Integral
Let \( i=(i_{1},\dots,i_{m}) \) be a multi-index consisting of the integers from \( 1 \) to \( n \), with \( {|i|=i_{1}+\dots+i_{n}} \); while \( \xi^{i}=\prod_{k=1}^{m}\xi_{ik} \), \( \xi=(\xi_{1},\dots,\xi_{n}) \), and
where \( (B_{\gamma_{i_{k}}})_{x_{i_{k}}}=\frac{\partial^{2}}{\partial x_{i_{k}}^{2}}+\frac{\gamma_{i_{k}}}{x_{i_{k}}}\frac{\partial}{\partial x_{i_{k}}} \) is a Bessel operator for all \( k=1,\dots,m \). Given an integer \( \alpha\geq 0 \), define the weighted Dirichlet integral of order \( \alpha \) as
In the image of (1), we obtain
Formula (6) can be used for defining the weighted Dirichlet integral \( d_{\alpha}^{\gamma} \) for an arbitrary \( \alpha\geq 0 \).
Lemma 1
Given \( 0<\alpha<1/2 \) and \( u\in{\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \), we have
where
Proof
If
then
Passing to the coordinates
we infer that
where \( \{\widetilde{y}_{2i-1}-x_{i}=z_{2i-1},\ i=1,\dots,n\} \) and \( \widetilde{}^{2n}_{+}=\{\widetilde{y}\in ^{2n}:{\widetilde{y}}_{2i}>0,\ i=1,\dots,n\} \). Putting \( z_{2i-1}=y_{i}\cos\beta_{i} \) and \( \widetilde{y}_{2i}=y_{i}\sin\beta_{i} \), with \( i=1,\dots,n \), we obtain
Therefore,
If \( 0<\alpha<1/2 \); then, using Parseval’s identity for (2) and formula 3.170 in [11, p. 155], we get
where
Since
where
we see that \( \mathcal{A}(\xi) \) is a homogeneous function of order \( 4\alpha \) which is invariant under orthogonal transformations and
Let us calculate \( C(n,\gamma,\alpha) \). Using the representation for \( \mathbf{j}_{\gamma} \) of the form \( \mathbf{j}_{\gamma}(x,\xi)=\mathbf{P}_{\xi}^{\gamma}\big{[}e^{-i\langle{}x,\xi\rangle{}}\big{]} \) (see [11, p. 137, formula 3.138]), where \( \mathbf{P}_{\xi}^{\gamma} \) is as in (5), we infer
Introduce the new coordinates in the last integral on the right-hand side as follows:
In these coordinates,
where \( \widetilde{x}=(\widetilde{x}_{1},\dots,\widetilde{x}_{2n})\in ^{2n} \), \( \widetilde{x}_{2i}>0 \); \( i=\overline{1,n} \), \( {|\widetilde{x}|=|z|} \), \( \widetilde{\xi}^{\prime}=(\xi_{1},0,\xi_{2},0,\dots,\xi_{n},0)\in ^{2n} \), \( {|\widetilde{\xi}^{\prime}|=|\xi|} \), and \( \widetilde{}^{2n}_{+}=\{x\in ^{2n}:{x}_{2i}>0,\ i=1,\dots,n\} \). As the integrand in \( C(n,\gamma,\alpha) \), we have some function of the type of a planar wave. Choosing integration over \( \widetilde{x}_{1}=p \), noticing that \( \big{\langle{}}\widetilde{x},\frac{\widetilde{\xi}^{\prime}}{|\xi|}\big{\rangle{}}=p \), and putting \( \widetilde{x\,}^{\prime}=(\widetilde{x}_{2},\dots,\widetilde{x}_{2n}) \), we obtain
Since
we have
for \( t=\rho\theta \). Here \( \widetilde{S}_{1}^{+}(2n-1) \) is the part of the unit sphere centered at the origin in \( ^{2n-1} \) for \( \widetilde{x}_{2i}>0 \), with \( i=1,\dots,n \). Applying formula (107) in [11, p. 49], we set
Consequently,
Finally, using Legendre’s duplication formula and Euler’s reflection formula, simplify \( C(n,\gamma,\alpha) \) to see that
Taking into account (8), (10), and the form of the constant \( C(n,\gamma,\alpha) \), we complete the proof of the lemma. ☐
The constant \( C(n,\gamma,\alpha) \) possesses the properties
Therefore,
in the remaining cases,
where \( l=[\alpha] \). The weighted integral \( d_{\alpha,\gamma} \) is continuous in \( \alpha \) and is independent of the orthogonal coordinates in \( ^{n}_{+} \).
4. Finding a Suitable Norm for \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \)
From the practical and theoretical standpoints, it is important to find out in what space and with what norm \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) is dense.
In [3], there was introduced some function space that is connected with multiplication by \( |x|^{-\alpha} \) in the images of the Hankel transform. This was called the space of Riesz \( B \)-potentials. Recall that in the theory of \( B \)-potentials a Riesz \( B \)-potential has the form (see [4])
An analog of Sobolev’s Theorem holds for \( U^{\alpha}_{\gamma} \) (see [4, Theorem 1]). Namely, for \( 0<\alpha<\frac{n+|\gamma|}{p} \), \( p>1 \), the operator \( U^{\alpha}_{\gamma} \) with density \( f\in L^{\gamma}_{p} \) is bounded from \( L_{p}^{\gamma} \) into \( L_{q}^{\gamma} \), where \( \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n+|\gamma|} \). For \( \alpha\geq\frac{n+|\gamma|}{p} \), the potential \( U^{\alpha}_{\gamma} \) can be defined in the sense of weighted distributions. As a consequence of this fact, in [3, Theorem 5], \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) was shown to be dense in the space of Riesz \( B \)-potentials only for \( 0<\alpha<\frac{n+|\gamma|}{p} \). Therefore, it is inconvenient to use the norm that bases on a Riesz \( B \)-potential in differential problems since these problems need potentials of arbitrarily large order.
We will prove that since norm convergence does not imply the pointwise convergence of a sequence in the spaces with the norm \( \sqrt{d_{\alpha,\gamma}} \); therefore, \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) is not a function space for \( \alpha\geq\frac{n+|\gamma|}{4} \).
Theorem 1
If \( \alpha\geq\frac{n+|\gamma|}{4} \) then the space \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) normalized by \( \sqrt{d_{\alpha,\gamma}} \) is not a function space with respect to any exceptional class.
Proof
Let \( u\in{\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) and let \( u \) be identically 1 in a neighborhood of zero in \( ^{n}_{+} \). Put \( u_{\rho}=u(x/\rho) \). We have
Consequently, for \( \alpha>\frac{n+|\gamma|}{4} \), we see that
This shows that the space under consideration cannot be a function space (since otherwise the whole of \( ^{n}_{+} \) must be an exceptional set).
Now, consider the case of \( \alpha=\frac{n+|\gamma|}{4} \). Choosing \( \varepsilon\in(0,\alpha) \) and \( v\in{\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \), for the bilinear norm corresponding to the quadratic form \( d_{\alpha,\gamma}(u) \), we see that
Here we used the Cauchy–Bunyakovsky inequality. Since
the equality \( \lim_{\rho\rightarrow\infty}d_{\alpha,\gamma}(u_{\rho},v)=0 \) is fulfilled for every \( v\in{\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \). We consider the Hilbert space that is a complete metric space with respect to the distance function induced by the inner product \( d_{\alpha,\gamma}(u,v) \). This space is the abstract completion of \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) under the norm \( \sqrt{d_{\alpha,\gamma}} \). Thus, \( u_{\rho} \) converges weakly to \( 0 \) as \( \rho\rightarrow\infty \) in this Hilbert space because \( d_{\alpha,\gamma}(u_{\rho}) \) is bounded. Then there exists a sequence \( \rho_{k}\rightarrow\infty \) such that the arithmetic means of \( \{u_{\rho_{k}}\} \) converges strongly to \( 0 \) (see [12]). But the equality \( \lim_{\rho\rightarrow\infty}u_{\rho}(x)=1 \) implies that the sequence of the arithmetic means pointwise converges everywhere to \( 1 \). Thus, \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) normalized by \( \sqrt{d_{\alpha,\gamma}} \) cannot be a function space. ☐
One of the simplest norms on \( {\overset{\circ}{C}}\,^{\infty}_{ev}(^{n}_{+}) \) equivalent to \( \sqrt{d_{\alpha,\gamma}} \) looks as
Below we prove that (11) is representable by the convolution kernels generating a generalized Bessel potential.
5. The Class of Generalized Bessel Potentials
In this section, we obtain a special representation of (11) which is most convenient for classes of generalized Bessel potentials. The generalization of the spaces of Bessel potentials has a rich history. In [13], for functions in the space of generalized Bessel potentials which is constructed from rearrangement invariant spaces, some equivalent description was obtained for the cone of decreasing rearrangements. In [14], some equivalent characterizations were established for the cones of decreasing rearrangements for the spaces of generalized Riesz and Bessel potentials.
A generalized Bessel potential is defined by the relation (see [15, 16])
where
is a generalization of the Bessel kernel. The two forms of the inverse operator of (12) were constructed in [16].
In [3], the space \( \mathbf{B}^{\alpha}_{\gamma}(L_{p}^{\gamma})=\{u:u=\mathbf{G}^{\alpha}_{\gamma}\varphi,\varphi\in L_{p}^{\gamma}\} \) with the norm \( \|u\|_{\mathbf{B}^{\alpha}_{\gamma}(L_{p}^{\gamma})}=\|\varphi\|_{L_{p}^{\gamma}} \) was introduced by using \( B \)-hypersingular integrals.
It was shown in [15] that
where \( K_{\frac{n+|\gamma|-\alpha}{2}} \) is a modified Bessel function of the second kind (see [10]).
Since \( G_{\alpha}^{\gamma} \) is integrable with the weight \( x^{\gamma} \) (see [15, 16]), its Hankel transform exists for every \( \xi \). The kernel \( G_{\alpha}^{\gamma} \) is an analytic function of \( \alpha \) for \( \alpha>0 \). Thus, from (13), by analytic continuation, we conclude that the Hankel transform of the generalized Bessel kernel for \( \alpha>0 \) is equal to
Moreover, the kernel \( G_{\alpha}^{\gamma} \) satisfies the properties
(see [15, 16]), where \( (G_{\alpha}^{\gamma}*G_{\beta}^{\gamma})_{\gamma} \) is the generalized convolution; see (4).
To obtain (11) for \( 0<\alpha<1/2 \), we first introduce the function
Then the generalized Bessel potential in (12) is representable as the generalized convolution operator of (4):
Below we will need \( \omega_{-4\alpha,\gamma}(|x|) \) for \( 0<\alpha<1/2 \). The asymptotic properties of the modified Bessel function \( K_{\nu} \) guarantee that the kernel function \( \omega_{-4\alpha,\gamma}(|x|) \) decreases exponentially at infinity and turns into a constant at the origin:
Consequently,
Theorem 2
The norm \( \|u\|_{\alpha,\gamma} \) admits the representation
Proof
Let \( \gamma_{n+1}\geq 0 \) be arbitrary and
where
and \( J_{\nu} \) is the Bessel function of the first kind. Acting as in proving (8) and applying (2) to the integral over \( y \), we infer
Putting \( (x,z_{0})=\widetilde{z} \), \( \widetilde{\xi}=(\xi,1) \), and \( \gamma^{\prime}=(\gamma,\gamma_{n+1}) \), we have
Performing the change of variables \( \widetilde{z}=z/|\widetilde{\xi}| \), we obtain
and
It follows from (9) that
and
where
Hence, for every \( \gamma_{n+1}\geq 0 \), we infer
Passing to \( \gamma_{n+1}=0 \) and putting
we write
Using the Wolfram Mathematica and (17), we get
Since
we have
Thus, we have validated (18). ☐
6. Conclusion
In conclusion, we observe that the two approaches are possible to defining the class of generalized Bessel potentials \( \mathbf{B}^{\alpha}_{\gamma}(L_{p}^{\gamma}) \) of order \( \alpha \) in \( ^{n}_{+} \). The first is that \( u\in\mathbf{B}^{\alpha}_{\gamma}(L_{p}^{\gamma}) \) if \( u \) is a generalized convolution \( (G_{\alpha}^{\gamma}*\varphi)_{\gamma} \) for some \( \varphi\in L^{\gamma}_{p}(^{n}_{+}) \). This approach was presented in [3], which used B-hypersingular integrals. The second approach consists in endowing \( \mathbf{B}^{\alpha}_{\gamma}(L_{p}^{\gamma}) \) with the norm
which can be written down on using the convolution kernel that generates the generalized Bessel potential. This expression shows that the quadratic interpolation between \( \|u\|_{\alpha,\gamma} \) and \( \|u\|_{\beta,\gamma} \) gives \( \|u\|_{\delta,\gamma} \), where \( \delta \) is the interpolation order \( \alpha(l-t)+\beta t \). The norm \( \|u\|_{\alpha,\gamma} \) is most convenient for studying the class of generalized Bessel potentials in \( ^{n}_{+} \).
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The first author was supported by the Ministry of Science and Higher Education of the Russian Federation (Project FEGS–2020–0001).
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Translated from Vladikavkazskii Matematicheskii Zhurnal, 2022, Vol. 24, No. 3, pp. 62–77. https://doi.org/10.46698/c3174-5520-8062-f
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Dzhabrailov, A.L., Shishkina, E.L. On the Theory of Spaces of Generalized Bessel Potentials. Sib Math J 64, 968–981 (2023). https://doi.org/10.1134/S0037446623040183
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DOI: https://doi.org/10.1134/S0037446623040183