Q: General strategy for studying the decay of eigenvalues of kernel integral operators Disclaimer. Please, be patient, I'm here to learn functional analysis...

Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite kernel $K:X \times X \rightarrow \mathbb R$, which is $L^2$ w.r.t the measure $\sigma \otimes \sigma$ on $X \times X$. Let  $L^2(X,\sigma)$ be the Hilbert space of all $\sigma$-measurable functions which are square-integrable w.r.t the measure $\sigma$, and consider the kernel integral operator induced by $K$, namely $I_K: L^2(X,\sigma) \to L^2(X,\sigma)$ be the associated kernel integral operator, defined by
$$
(I_K f)(x) := \int_X K(x,y)f(y)d\sigma(y).
$$
Note that $I_K$ is compact and self-adjoint operator (to check!) on the nontrivial Hilbert space $L^2(X,\sigma)$. Consequently, it has only countably many eigenvalues (including multiplicities) $\lambda_1 \ge \lambda_2 \ge \ldots \lambda_n \ge \ldots$. Moreover, $\lambda_n \to 0$ as $n \to \infty$.

Question. Is there a general strategy (tools, theorems, etc.) for studying the rate of decay of $\lambda_n$ as a function of the regularity properties (Lipschitz, Hoelder class, etc.) of the input kernel $K$ ?

Note. I'm particularly interested in kernels of the form $K(x,y) \equiv k(x^Ty)$, for some $k:[-1,1] \to \mathbb R$.

A: Birman and Solomyak have studied this question quite intensivly.
The paper may not be the easiest to understand, but it does cover in a very general setup, what regularity conditions on the kernel imply in terms of singular number estimates. As you are working with a positive kernel, those are equivalent to the eigenvalues.
If you identify the sphere with a cube $[0,1]^{m-1}$ and transform the measure in a suitable way, you can apply Propositions 2.1-2.3. The technical conditions are a bit more complicated then Hölder continuity, and are defined in section 1.
doi link to the paper
Let $W^{\alpha,p}(X)$ be the fractional Sobolev space with $\alpha$ degrees of differentiability and $p$-integrable functions. Let $W^{\alpha,p}_{hom}(X)$ be the associated homogenuous space (we only care for the highest differential and the function does not need to be $p$-integrable itself).

(2.14): $1- r^{-1} = 2(p^{-1} - \alpha m^{-1})$


To simplify the statement of the theorem
we use the notation $D^\alpha_pX$ in the following sense: for $p\alpha > m$ we take $X = Q^m$ [the unit cube] and $D^\alpha_pX= W^{\alpha,p} Q^m$ ; for $p\alpha  < m$ we take $X = \mathbb R^m$ and $D^p_\alpha X =W_{hom}^{\alpha,p}(X)$.
. The notation $D^{\alpha_1}_{p_1}(X) D^{\alpha_2}_{p_2}(Y) $ is to be interpreted similarly (four cases are possible here).
THEOREM 2.5  Let $X = Q^{m_1}$ or $X = \mathbb R^{m_1}$, $Y = Q^{m_2}$ or $Y = \mathbb R^{m_2}$ and $T \in D^{\alpha_1}_{p_1}(X) D^{\alpha_2}_{p_2}(Y)$ for $2 \le p_1$ and $p_2 < \infty$. Let $\rho \in M_{r_1}(X)$ and $\tau \in M_{r_2}(Y)$ with $r_i= 1$ for $p_i\alpha_i > m_i$
and $r_i > 1$ for $ρ_i\alpha_i = m_i$
and suppose that
(2.14) is satisfied when $ρ_i \alpha_ i < m_i$($i=1, 2$). Then
$$s_n(T_{\rho \tau}) \le C n^{-\gamma} N(T \mid D^{\alpha_1}_{p_1}(X)D^{\alpha_2}_{p_2}(Y) )  N^{\frac 1 2} ( \rho \mid M_{r_1}(X) )  N^{\frac 1 2} ( \tau \mid M_{r_2}(Y) ) ,
$$
$$ \gamma = \frac 1 2+ \frac {\alpha_1} {m_1} + \frac {\alpha_2} {m_2}. $$

Here $M_r(X)$ is the space of all measures, such that the density with respect to the Lesbegue measure is in $L^r$. The operator $T_{\rho \tau}$ is the integral operator with kernel $T$ from $L^2(Y,\tau)$ to $L^2(X,\rho)$ and the norm in the composition of spaces $H( X) H'( Y)$ is the space of all kernels, such that the map from $X$ to the Banach space $H'(Y)$ is in the space $H(X)$.
I am not sure how understandable that is. As I said, the paper is not easy to understand and even the statement is quite hard to tell.
Different idea:
Another possible approach would be to consider the Laplace operator on $S^d$ and to show that it satisfies a Weyl law. Then you could use that the operator $(- \Delta)^{-\frac s 2} $ has decaying eigenvalues and is in some $p$ Schatten class (meaning p-summability of the eigenvalues). Than you can conclude that if $(-\Delta)^{\frac s 2 }K$ is in the Hilbert Schmidt class (square integrable), the operator $T$ itself can be written as a comoposition of a $p$-Schatten class operator and a HS operator. Hence, it is in the $Q$-Schatten class for $\frac 1 q= \frac 1 2 + \frac 1 p$ by Hölder.
