Summary of PhD thesis

István Blahota

In my dissertation I deal with the diadic-harmonic analysis, particularly with the Vilenkin and Vilenkin-like systems. In what follows I will use the usual notations of the Vilenkin analysis.

Let A_{n} be the σ-algebra generated by the cosets I_{n}(z), where n ∈ ℕ, z ∈ G_{m}. Let α_{j}^{k}, α_{n} (k, j, n ∈ ℕ) be functions satisfying the following conditions:

(i) α_{j}^{k} : G_{m} → ℂ is A_{j} -measurable (k, j ∈ ℕ), (ii) |α_{j}^{k}| := α_{0}^{k} := α_{j}^{0} := α_{j}^{k}(0) := 1 (k, j ∈ ℕ),

(iii) α_{n} := ∏_{j=0}^{∞} α_{j}^{n^{(j)}} (n ∈ ℕ, n^{(j)} := ∑_{k=j}^{∞} n_{k}M_{k}).

Let χ_{n} = ψ_{n}α_{n} (n ∈ ℕ). A function system {χ_{n} : n ∈ ℕ} of this type is called a ψα system on the Vilenkin group G_{m}.

It is proved that this system is orthonormal and complete in L^{1}(G_{m}). I proved several generalizations with respect to the ψα and to the double

ψα system.

Let k ∈ ℙ. We say that f : ℕ → ℂ is an arithmetical function even modulo k, if every n ∈ ℕ

f(n) = f((n, k)),

where (n, k) is the greatest common divisor of n and k.

Let

B_{k} := {f even modulo k}, B := ⋃_{k∈N} B_{k}.

If f : N → R then

\overline{M}(f) := lim sup_{x→∞} x^{−1} ∑_{1≤j≤x} f(j)

are called the upper mean values of f.

Let 1 ≤ q < ∞. Then the upper mean value gives rise to a seminorm

∥f∥_{q} := \{\overline{M}(|f|^{q})\}^{1/q}

on the linear space {f|f : N → C, ∥f∥^{q} < ∞}. We denote by B^{q} the closure of B in this norm. Functions in B^{q} are called almost even arithmetical functions.

If n ∈ ℕ, n = ∑_{j=0}^{∞} n_{j}M_{j}, then let

n̆ := ∑_{j=0}^{∞} n_{j}M_{j+1}^{−1}

and of course only a finite number of n′_{j}s is not zero.

Denote by

κ_{n}(x) := lim_{k→∞} exp(2πın̆ ∑_{j=0}^{k} x_{j}M_{j}) (n ∈ ℕ, x ∈ G_{m}).

There exists the limit, and the system {κ_{n}|n ∈ ℕ} on G_{m} is a ψα system. If f ∈ B^{q} and 1 ≤ q < ∞, then there exists f^{∗} ∈ L^{q}(G_{m}) μ almost

everywhere unique, that f^{∗}(λ^{−1}(n̆)) = f(n), ∥f^{∗}∥_{q} = ∥f∥_{B^{q}} and f_{n} → f ⇔ f_{n}^{∗} → f^{∗}, where λ is the so-called Fine-map.

If for every k ∈ ℙ there exists n = n(k) such that k| M_{n}, we say that G_{m} is a R-Vilenkin group.

Theorem If G_{m} is a R-Vilenkin group, then there exists F function which is an element of B^{q} where 1 ≤ q, the Vilenkin—Fourier series of which diverges everywhere.

My main results are the following (both for bounded and double systems): Theorem The operator σ^{∗} is of type (H^{∘}, L) i.e.

∥σ^{∗}f∥_{1} ≤ c∥f∥_{H^{∘}} (f ∈ H).

Theorem Let f ∈ L^{1}(G_{m} ×Gm̃). Then σ_{(n_{1},n_{2})}f → f a.e. as n_{1}, n_{2} → ∞ , where n_{1}, n_{2} ∈ ℙ and β^{−1} ≤ n_{1}/n_{2} ≤ β for some fixed parameter β ≥ 1.

The other system (also introduced by György Gát) is a common generalization of several well-known systems.

Let us introduce this one on G_{m}, which will be called Vilenkin-like system. Now G_{m} is not the Vilenkin group, but the so-called Vilenkin space without any operator.

The complex valued functions r_{k}^{n} : G_{m} → ℂ (k, n ∈ ℕ) we call generalized Rademacher functions, if they have the following properties:

(i.) r_{k}^{n} is A_{k+1} measurable (i.e. r_{k}^{n}(x) depends only on x_{0}, ..., x_{k} (x ∈ G_{m})), r_{k}^{0} = 1 for all k, n ∈ ℕ.

(ii.) If M_{k} is a divisor of n, l and n^{(k+1)} = l^{(k+1)} (k, l, n ∈ ℕ), then

E_{k}(r_{k}^{n}r̄_{k}^{l}) = \begin{cases}1 if n_{k} = l_{k}, \\ 0 if n_{k} ≠ l_{k} \end{cases}

(z̄ is the complex conjugate of z).

(iii.) If M_{k} is a divisor of n (that is, n = n_{k}M_{k}+n_{k+1}M_{k+1}+...+n_{|n|}M_{|n|}). Then

∑_{n_{k}=0}^{m_{k}−1} |r_{k}^{n}(x)|^{2} = m_{k}

for all x ∈ G_{m} Now define the Vilenkin-like system ψ := (ψ_{n} : n ∈ ℕ) as follows

ψ_{n} := ∏_{k=0}^{∞} r_{k}^{n^{(k)}}, n ∈ ℕ.

(Since r_{k}^{0} = 1, then ψ_{n} := ∏_{k=0}^{|n|} r_{k}^{n^{(k)}}). The Vilenkin-like system ψ is orthonormal.

I proved a norm inequality with respect to this system. Theorem There exists a c > 0 absolute constant that if f ∈ H(G_{m}) and

the sequence m is bounded, then

sup_{N∈ℙ} \frac{1}{log N} ∑_{n=1}^{N} \frac{∥S_{n}f∥_{1}}{n} ≤ c∥f∥_{H}.

It is managed to verify the two-dimensional version of this inequality. Theorem Assume that β ≥ 1, f ∈ H^{1}(G_{m} × G_{m}) and 2 ≤ N, K ∈ ℙ.

Then there exists a constant c > 0 depending only on β such that

sup_{N,K} \frac{1}{log N log K} ∑_{β^{−1}≤n/k≤β(n,k)≤(N,K)} \frac{∥S_{n,k}f∥_{1}}{nk} ≤ c∥f∥_{H^{1}}.