Some new estimates for generalized fractional integrals associated with operators on Morrey spaces
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2026
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| _version_ | 1866913144614420480 |
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| author | Wang, Hua |
| author_facet | Wang, Hua |
| contents | Let $\mathcal{L}$ be the infinitesimal generator of an analytic semigroup $\big\{e^{-t\mathcal L}:t>0\big\}$ on $L^2(\mathbb R^n)$ with Gaussian upper bounds, and suppose that $\mathcal{L}$ has a bounded holomorphic functional calculus on $L^2(\mathbb R^n)$. For given $0<α<n$, let $\mathcal L^{-α/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is given by \begin{equation*} \mathcal L^{-α/2}(f)(x):=\frac{1}{Γ(α/2)}\int_0^{+\infty}e^{-t\mathcal L}(f)(x)t^{α/2-1}dt, \end{equation*} where $Γ(\cdot)$ is the usual gamma function. In the limiting Sobolev case $λ=n-αp$ and $1\leq p<n/α$, the author proves that the operator $\mathcal{L}^{-α/2}$ is bounded from the Morrey space $M^{p,λ}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$, and is bounded from the vanishing Morrey space $VM^{p,λ}(\mathbb R^n)$ into $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$, where $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ and $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$ are the spaces of bounded mean oscillation and vanishing mean oscillation associated with the operator $\mathcal{L}$, respectively. As a consequence, the author obtains that the operator $\mathcal{L}^{-α/2}$ is bounded from $L^{p,\infty}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ when $p=n/α$ and $0<α<n$. The proofs are based on pointwise kernel estimates of the operators $\mathcal L^{-α/2}$ and $(I-e^{-t\mathcal L})\mathcal{L}^{-α/2}$ for $0<α<n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_19372 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some new estimates for generalized fractional integrals associated with operators on Morrey spaces Wang, Hua Classical Analysis and ODEs 42B20, 42B35, 46E30, 47G10 Let $\mathcal{L}$ be the infinitesimal generator of an analytic semigroup $\big\{e^{-t\mathcal L}:t>0\big\}$ on $L^2(\mathbb R^n)$ with Gaussian upper bounds, and suppose that $\mathcal{L}$ has a bounded holomorphic functional calculus on $L^2(\mathbb R^n)$. For given $0<α<n$, let $\mathcal L^{-α/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is given by \begin{equation*} \mathcal L^{-α/2}(f)(x):=\frac{1}{Γ(α/2)}\int_0^{+\infty}e^{-t\mathcal L}(f)(x)t^{α/2-1}dt, \end{equation*} where $Γ(\cdot)$ is the usual gamma function. In the limiting Sobolev case $λ=n-αp$ and $1\leq p<n/α$, the author proves that the operator $\mathcal{L}^{-α/2}$ is bounded from the Morrey space $M^{p,λ}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$, and is bounded from the vanishing Morrey space $VM^{p,λ}(\mathbb R^n)$ into $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$, where $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ and $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$ are the spaces of bounded mean oscillation and vanishing mean oscillation associated with the operator $\mathcal{L}$, respectively. As a consequence, the author obtains that the operator $\mathcal{L}^{-α/2}$ is bounded from $L^{p,\infty}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ when $p=n/α$ and $0<α<n$. The proofs are based on pointwise kernel estimates of the operators $\mathcal L^{-α/2}$ and $(I-e^{-t\mathcal L})\mathcal{L}^{-α/2}$ for $0<α<n$. |
| title | Some new estimates for generalized fractional integrals associated with operators on Morrey spaces |
| topic | Classical Analysis and ODEs 42B20, 42B35, 46E30, 47G10 |
| url | https://arxiv.org/abs/2605.19372 |