Boundedness of Riesz Transform on (subsets of) Hölder spaces?

Question

Definition and setup The Riesz transform for say $C^\infty_c(\mathbb R^d)$ functions $f$ is defined by a principal value integral, $$ Rf(x) := c_d \operatorname{pv}\!\!\!\int_{\mathbb R^d} \frac{y}{|y|^{d+1}}f(x-y) \, dy := c_d \lim_{\epsilon\downarrow 0} \int_{|y|>\epsilon} \frac{y}{|y|^{d+1}}f(x-y) \, dy,$$ The integral is interpreted componentwise, the constant $c_d$ is chosen so that the Fourier transform $\int_{\mathbb R^d} Rf(x)e^{-2\pi i x\xi} \, dx = \frac{- i\xi}{|\xi|}. $ Wikipedia link.

It is well-known that the Riesz transform is bounded on $L^p$ spaces, $p\in(1,\infty)$, and commutes with derivatives. Its therefore bounded on Sobolev spaces $W^{s,p}$, $p\in(1,\infty)$.

Question I think I've seen before that the Riesz transform is bounded as a map $C^\alpha \cap L^p \to C^\alpha \cap L^p$? Is this true? A paper I've read casually remarked that the Riesz transform is bounded on Hölder and Sobolev Spaces, and I presume this is the kind of result they mean.

I gave it a few naive tries. I can show that under the assumption that $f\in L^p \cap C^\alpha, p\in[1,\infty)$, the integral form is well-defined and $Rf\in L^\infty$, with for any $\lambda>0$ $$ |R_jf(x)| \lesssim_d [f]_\alpha \lambda^\alpha + \|f\|_{L^p} \lambda^{-d/p}$$ or if you try to minimise in $\lambda$ you get something like $\|Rf\|_{L^\infty} \lesssim_d [f]_\alpha^{\frac{d/p}{\alpha+d/p}}\|f\|_{L^p}^{{\frac{\alpha}{\alpha+d/p}}}$. But I feel like I'm missing a "standard trick" to continue to estimate $[Rf]_\alpha$ (in particular I don't know how to use the cancellation), and revisiting some books like Stein's, I couldn't find the result or the trick I feel I need. Any pointers?

Update A friend has pointed out that it is in Stein's book, in the form of a "Further Result" (i.e. exercise). It is 6.9 on pages 50-51. He gives the hint that if the Kernel $\frac{\Omega(y)}{|y|^d}$ (in our case $\Omega(y) = y/|y|$) is sufficiently smooth then the proof is "elementary" (NB the quotation marks are Stein's), and directs the reader to 3 references:

1. J. Privalov "Sur Les fonctions conjuguées," Mat. Zeit. 26 (1927), 218-244.
2. A. P. Calderón and A. Zygmund, "Singular Integrals and Periodic Functions," Studia Math. 14 (1954), 249-271.
3. M. H. Taibleson, "The preservation of Lipschitz spaces under singular integral operators," Studia Math. 24 (1963), 105-111.

So it might be possible to distill an answer from one of these papers...

Answer 1

Let's pretend $\newcommand{\pv}{\text{pv}\!\!\!}\newcommand{\kernel}[1]{\frac{#1}{{|#1|}^{d+1}}}c_d=1$. (It seems that there is no use for the $L^p$ control when estimating the seminorm? Would appreciate a check.) \begin{align} Rf(x) &= \quad\pv\int\kernel{x-y}f(y)dy = \pv\int\kernel{x-y}(f(y)-f(x))dy. \\ Rf(x+h)-Rf(x)&=\quad\pv\int \kernel{x+h-y}(f(y)-f(x+h))dy \\ &\quad - \pv\int \kernel{x-y}(f(y)-f(x))dy \\ &= \quad\pv\int_{|x-y|<2|h|} \kernel{x+h-y}(f(y)-f(x+h))dy \\& \quad - \pv\int_{|x-y|<2|h|} \kernel{x-y}(f(y)-f(x))dy \\& \quad - \pv\int_{|x-y|\ge 2|h|} \kernel{x-y}(f(x+h)-f(x))dy \\& \quad - \pv\int_{|x-y|\ge 2|h|} \left[\kernel{x-y}-\kernel{x+h-y}\right](f(y)-f(x+h))dy \\&= I_1+I_2+I_3+I_4. \end{align} $I_3=0$ by the mean value zero of the kernel.

For $I_1$, you use the $C^\alpha$ regularity: $$|I_1|\le \int_{|x+h-y|<3|h|}\frac1{|x+h-y|^{d-\alpha}} [f]_\alpha dy = C_{d,\alpha} [f]_\alpha |h|^\alpha $$ The factor $|h|^\alpha$ easily computed using spherical coordinates. Similarly with $I_2$.

As for $I_4$, mean value theorem on the kernel gives (noting that $|x-y|>2|h|$ so that $|x+h-y|>|h|$ and therefore the line segment $[x-y,x+h-y]$ always avoids the origin) $$ \left|\kernel{x-y}-\kernel{x+h-y}\right| \le |h| \sup_{z\in[x-y,x+h-y]}\nabla_z\left( \kernel{z}\right)\lesssim_d \frac{|h|}{|x-y|^{d+1}}.$$ Thus (since $x^\alpha$ is subadditive) \begin{align}|I_4| &\lesssim_d \int_{|x-y|>2|h|}\frac{|h|[f]_\alpha |x+h-y|^\alpha}{|x-y|^{d+1}} dy \\ &\le \int_{|x-y|>2|h|}\frac{|h|^{1+\alpha} [f]_\alpha }{|x-y|^{d+1}} dy+\int_{|x-y|>2|h|}\frac{|h|[f]_\alpha }{|x-y|^{d+1-\alpha}} dy \\ &\lesssim_{d,\alpha} [f]_\alpha h^\alpha. \end{align} Finally, $$ [Rf]_{\alpha} \lesssim_{d,\alpha} [f]_{\alpha},$$ which proves: $$ \|Rf\|_{L^p\cap C^\alpha} \lesssim_{d,\alpha} \|f\|_{L^p\cap C^\alpha}.$$ (The norm on the intersection of Banach spaces is $\|f\|_{X\cap Y} = \|f\|_X+\|f\|_Y$.)