# Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order

I would like to show the following function is $$\gamma$$-Hölder continuous. Said function $$F:\mathbb{R}^n \rightarrow \mathbb{R}$$ is defined by a singular integral operator of convolution type as follows (for some fixed $$i$$ and $$j$$)

$$F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) \right)f(y)dy$$

where $$f \in C_c^{\gamma}(\mathbb{R}^n)$$ (ie a compactly supported $$\gamma$$-Hölder continuous function) and the integral kernel $$k:\mathbb{R}^n\backslash \lbrace \vec{0} \rbrace \rightarrow \mathbb{R}^n$$ is defined as

$$\begin{equation} k_i(x) =\frac{x_i}{|x|^n} \end{equation}$$

Note $$k$$ is the derivative of the Newtonian kernel up to a constant factor. In addition, we have $$Y: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ such that

$$\| \nabla Y \|_{1, \gamma} = |Y(0)|+\|\nabla Y\|_0 + |\nabla Y|_{1,\gamma} < \infty$$ where $$|W|_{\gamma} = \sup_{x \neq y} \frac{|W(x)-W(y)|}{|x-y|^{\gamma}}$$ is the $$\gamma$$-Hölder seminorm. Note $$Y$$ has bounded $$\gamma$$-Hölder continuous derivatives.

-----------------------------Attempt at Solution -------------------------------------

In attempting to prove the $$\gamma$$-Holder continuity of $$F$$, my strategy thus far has involved expanding $$Y$$ as a first order Taylor polynomial as follows $$Y_k(y)=Y_k(x) + \partial_l Y_k(x)(y_l-x_l) +(y_l-x_l)\int_0^1 \left( \partial_l Y_k(x+t(y-x)) -\partial_l Y_k(x) \right)dt$$ I'll denote the remainder term as $$R_x(y)$$. Note $$|R_{x}(y)| \leq C|y-x|^{1+\gamma}$$.

So I write $$\begin{eqnarray} && F(x+h)-F(x)=\lim_{\epsilon \rightarrow 0} \bigg( \int_{|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)f(y)dy \\ &&- \ \int_{|x-y|\geq \epsilon} \partial_k \partial_j k_i(x-y)\partial_lY(x)(x_l+h_l-y_l)f(y)dy -\int_{|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lR_{x+h}(y)f(y)dy \\ && + \int_{|x-y|\geq \epsilon} \partial_k \partial_j k_i(x-y)\partial_lR_{x}(y)f(y)dy \bigg) \end{eqnarray}$$

I can show the absolute value of the difference of the first two terms is bounded by $$C|h|^{\gamma}$$. This is done by first observing that $$\partial_j k_i$$ is the integral kernel of a singular integral operator (ie it is smooth everywhere but $$\vec{0}$$, is homogeneous of degree $$-n$$, and has mean value on the unit sphere). In fact, this mean-value property extends as follows:

$$\begin{equation} \int_{|x|=1}x^{\beta}D^{\alpha}k_i(x)dS=0 \quad \Rightarrow \int_{r \leq |x|\leq R} x^{\beta}D^{\alpha}k_i(x) dx = 0 \end{equation}$$

for any multi-indices $$\alpha$$ and $$\beta$$ such that $$|\alpha|=|\beta|+1$$. In particular terms of the form

$$\begin{equation} C\int_{r \leq |x|\leq R} x_l \partial_k\partial_jk_i(x) dx = 0 \end{equation}$$ for C constant can be added freely.

The first two terms can be dealt with (roughly, this is jut a sketch) in two steps. The first is

$$\begin{eqnarray} && \int_{ |h|>|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)f(y)dy \\ && = \int_{ |h|>|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)\left(f(y)-f(x) \right) dy \leq c|f|_{\gamma}|h|^{\gamma} \end{eqnarray}$$

where we made use of the $$\gamma$$-Holder continuity of $$f$$ and the mean-value property of the kernel by adding a zero term. The second is

$$\begin{eqnarray} && \int_{ |x+h-y|\geq|h|,|x-y|\geq 2|h|} \Big( \partial_k \partial_j k_i(x+h-y)(x_l+h_l-y_l) - \partial_k \partial_j k_i(x-y)(x_l-y_l) \Big)\left(f(y)-f(x+h) \right) dy \\ && \leq c|f|_{\gamma}|h|^{\gamma} \end{eqnarray}$$

through the use of the mean-value theorem on the integral kernel $$\partial_k \partial_j k_i$$ and the further addition of zero terms.

--------------- Remaining Problems -----------------

I have trouble bounding the difference of the remainder terms by $$C|h|^{\gamma}$$. Following a similar method as above I can bound them by $$C|h|^{\xi}$$ for $$\xi < \gamma$$. But perhaps there is a more general theorem I could use? I'd also be interested in textbook recommendations covering similar subject matter.

This is covered in Gilbarg-Trudinger, and I think it can be put under the heading "potential theory approach to Schauder estimates". "Kellog's theorem" might reveal something too. It is closely related to the Calderon-Zygmund theory of singular integral operators, although the latter is typically concerned with $L^p$ spaces.